Methods and systems for solving a problem on a quantum computer

ABSTRACT

A method of solving a problem can include providing a fermionic Hamiltonian, transformation of the fermionic Hamiltonian to qubit operators, transformation of the fermionic Hamiltonian in qubit operators to a mean-field Hamiltonian, and embedding the Hamiltonian onto a quantum computer. Such systems and methods may improve upon existing methods for solving electronic structure problems on a computer by adapting the problem to available hardware, reducing computational cost, and reducing the number of required qubits to solve electronic structure problems for larger number of atoms.

CROSS-REFERENCE

This application claims the benefit of U.S. Provisional Application No.62/783,275, filed Dec. 21, 2018, U.S. Provisional Application No.62/729,748, filed Sep. 11, 2018, and U.S. Provisional Application No.62/678,936, filed May 31, 2018, which applications are incorporatedherein by reference in their entireties.

BACKGROUND

In molecular simulations, electrons, protons, and neutrons—all quantummechanical in nature—interact in many body interactions whose solutionmay be intractable using conventional computers and conventionalnumerical methods due to long computational times. Quantum computers maybe particularly suited to solving these problems. However, currentlyavailable quantum computers may have significant hardware limitations interms of number of qubits available, gate depth, and gatefidelity/accuracy, which may limit the use of such hardware. While itmay be possible to simulate larger molecules as quantum computingtechnology improves, even as greater numbers of qubits become availablethere will remain a need to simulate larger and larger molecules.

SUMMARY

Recognized herein is a need to improve computational accuracy and toreduce the number of qubits required to simulate a molecule of a givensize, thereby reducing the computational cost and improving the functionof the quantum computer. Systems and methods disclosed herein may reducethe number of quantum circuit gate operations (e.g. the circuit depth)to perform a calculation and, in some cases, may even reduce the numberof quantum circuit gate operations to achieve the full configurationinteraction equivalent energy. For example, since each quantum gateoperation may introduce noise into the calculation, reducing the numberof gate operations may reduce the final error. Systems and methodsdisclosed herein may modify the Hamiltonian to improve embedding on aquantum annealer

In an aspect, a system operable to solve a problem is provided. In someembodiments, the system comprises a quantum computer comprising aplurality of qubits; a qubit Hamiltonian, wherein one or morecoordinates in the qubit Hamiltonian comprises a parametrization in spincoherent states, wherein the parametrization comprises either anoperation of one or more quantum logic gates or an expression of thequbit Hamiltonian in Pauli Z rotations, wherein the Hamiltonian isembedded on the quantum computer, wherein one or more eigenvalues of thequbit Hamiltonian is a variational upper bound to an exact energy, andwherein a lower eigenvalue of the qubit Hamiltonian comprises a solutionto the problem.

In some embodiments, the system may comprise a spin coherent statecomprising an expression in spherical polar coordinates on the Blochsphere. In some embodiments, the system may comprise the operation ofone or more quantum logic gates comprising a qubit mean-field ansatz. Insome embodiments, the system may comprise a qubit Hamiltonian comprisingan Ising type Hamiltonian. In some embodiments, the system may comprisea qubit Hamiltonian comprising a quadratic unconstrained boundaryoptimization problem. In some embodiments, the system may comprise aqubit Hamiltonian in Pauli Z rotation comprising bias terms and couplingterms for each of the plurality of qubits. In some embodiments, thesystem may comprise a value of the bias terms and a value of thecoupling terms which is determined using a classical computer.

In some embodiments, the system may comprise a solution to the problembeing a quantum state of a molecule. In some embodiments, the solutionmay comprise a quantum state which is a ground state. In someembodiments, the molecule may be an organic optoelectronic material. Insome embodiments, the molecule may comprise one or more main groupelements. In some embodiments, the molecule may be a polymer. In someembodiments, the molecule may be a molecular crystal.

In some embodiments, the system may comprise a qubit Hamiltonian whichis a mean-field Hamiltonian. In some embodiments, the system maycomprise a qubit Hamiltonian comprising a transformation of thefermionic Hamiltonian. In some embodiments, the system may comprise atransformation of the fermionic Hamiltonian comprising using theJordan-Wigner transformation. In some embodiments, the system maycomprise a transformation of the fermionic Hamiltonian comprising usingthe Bravyi-Kitaev method.

In some embodiments, the system may comprise a quantum computer which isa quantum annealer. In some embodiments, the system may comprise a lowereigenvalue of the reduced Hamiltonian. The lower eigenvalue may be aglobal minimum. In some embodiments, the system may comprise a lowereigenvalue of the reduced Hamiltonian is a local minimum. In someembodiments, the system may comprise a quantum computer which is auniversal gate quantum computing unit.

In another aspect, a method of solving a problem on a quantum computeris provided. In some embodiments, the method may comprise: providing aqubit Hamiltonian, wherein one or more eigenvalues of the qubitHamiltonian is a variational upper bound to an exact state energy;parametrizing one or more coordinates in the Hamiltonian by spincoherent states, wherein parametrizing the one or more coordinatescomprises either an operation of one or more quantum logic gates orexpressing the qubit Hamiltonian in Pauli Z rotations; embedding thequbit Hamiltonian on the quantum computer; and determining a lowereigenvalue of the qubit Hamiltonian, wherein the lower eigenvaluecomprises a solution to the problem.

In some embodiments, the method may comprise a spin coherent statecomprising an expression in spherical polar coordinates on the Blochsphere. In some embodiments, the method may comprise an operation of oneor more quantum logic gates comprising a qubit mean-field ansatz. Insome embodiments, the method may comprise a qubit Hamiltonian comprisingan Ising type Hamiltonian. In some embodiments, the method may comprisea qubit Hamiltonian comprising a quadratic unconstrained boundaryoptimization problem. In some embodiments, the method may comprise aqubit Hamiltonian in Pauli Z rotation comprises bias terms and couplingterms. In some embodiments, the method may comprise determining a valueof the bias terms and the coupling terms using a classical computer. Insome embodiments, the method may comprise transferring the value of thebias terms and the value of the coupling terms to the quantum computer.

In some embodiments, the method may comprise a solution to the problemwhich is a quantum state of a molecule. In some embodiments, thesolution may comprise a quantum state is a ground state. In someembodiments, the molecule may comprise an organic optoelectronicmaterial. In some embodiments, the molecule may comprise one or moremain group elements. In some embodiments, the molecule may comprise apolymer. In some embodiments, the molecule may comprise a molecularcrystal.

In some embodiments, the method may comprise a qubit Hamiltonian whichis a qubit mean-field Hamiltonian. In some embodiments, the method maycomprise transforming a fermionic Hamiltonian to the qubit Hamiltonianusing a Jordan-Wigner transformation. In some embodiments, the methodmay comprise transforming a fermionic Hamiltonian to the qubitHamiltonian using the Bravyi-Kitaev method. In some embodiments, themethod may comprise transforming a fermionic Hamiltonian to the qubitHamiltonian using the Parity method. In some embodiments, the method maycomprise one or more gate operations which are used to determine thelower eigenvalue of the qubit Hamiltonian. In some embodiments, themethod may comprise a number of gate operations used which scaleslinearly with a number of fermions in the qubit Hamiltonian.

In some embodiments, the method may comprise a quantum computer which isa quantum annealer. In some embodiments, the method may comprise aquantum computer which is a universal gate quantum processing unit. Insome embodiments, the method may comprise a solution which is a globalminimum. In some embodiments, the method may comprise a solution whichis a local minimum. In some embodiments, the method may further compriseproviding the solution to a user

In an aspect, a method of solving a problem on a quantum computer isprovided. The method may comprise: providing a qubit Hamiltonian,wherein one or a plurality eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact state energy; providing a set ofentanglers, the entanglers comprising one or a plurality Pauli words;determining a subset of the set of entanglers which reduces a value ofthe one or plurality eigenvalues; truncating the set of entanglers;forming a qubit coupled cluster Hamiltonian comprising the truncated setof entanglers; embedding the qubit coupled cluster Hamiltonian on thequantum computer; and determining a lower eigenvalue of the qubitHamiltonian, wherein the lower eigenvalue comprises a solution to theproblem.

In some embodiments, the method may further comprise parameterizing oneor more coordinates of the Hamiltonian by spin coherent states. In someembodiments, the method may further comprise expressing the truncatedset of entanglers as Pauli z-rotations. In some embodiments, the methodmay further comprise folding an optimization domain in amplitude orphase space before determining the lower eigenvalue of the qubit coupledcluster Hamiltonian.

In another aspect, a quantum computer is provided. The quantum computermay comprise: a plurality of qubits; a Hamilton embedded on the quantumcomputer, wherein one or a plurality of eigenvalues of the Hamiltonianis a variational upper bound to an exact energy; and a plurality of gateoperations which may be implemented by the quantum computer, theplurality of gate operations comprising: a set of entanglers comprisingone or a plurality of Pauli Words; and a subset of entanglers, whereinthe subset of entanglers comprises selected Pauli Words which reduce avalue of the one or a plurality of eigenvalues.

In some embodiments, the Hamiltonian may comprise a qubit coupledcluster Hamiltonian. In some embodiments, the subset of entanglers maybe parametrized in the Hamiltonian by spin coherent states. In someembodiments, the subset of entanglers may be expressed as Pauliz-rotations. In some embodiments, a domain of optimization of the subsetof entanglers may be folded in amplitude or phase.

In an aspect, a method of solving a problem on a quantum computer isprovided. The method may comprise: providing a qubit Hamiltonian,wherein one or more eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact state energy; providing an operator,wherein the operator commutes with the qubit Hamiltonian; constrainingthe qubit Hamiltonian with respect to the operator and an eigenvalue ofthe operator; embedding the qubit Hamiltonian on the quantum computer;and determining a lower eigenvalue of the qubit Hamiltonian, wherein thelower eigenvalue comprises a solution to the problem.

In some embodiments, the operator may comprise a commutation relation ofthe qubit Hamiltonian and a second operator that commutes with the qubitHamiltonian. In some embodiments, the second operator may be a numberoperator. In some embodiments, the second operator may be a total spinoperator. In some embodiments, the operator may be a number operator. Insome embodiments, the operator may be a total spin operator. In someembodiments, the method may further comprise constraining the qubitHamiltonian with respect to a second operator and a second eigenvalue ofthe second operator and wherein the second operator commutes with thequbit Hamiltonian. In some embodiments, the operator may be generatedusing a Bravyi-Kitaev or a Jordan-Wigner transformation of a relatedfermionic operator. In some embodiments, the method further comprisesparametrizing one or more coordinates in the qubit Hamiltonian by spincoherent states, wherein parametrizing the one or more coordinatescomprises either an operation of one or more quantum logic gates orexpressing the qubit Hamiltonian in Pauli Z rotations.

In another aspect, a quantum computer is provided. The quantum computermay comprise: a plurality of qubits; a qubit Hamilton embedded on thequantum computer, wherein one or a plurality of eigenvalues of the qubitHamiltonian is a variational upper bound to an exact energy, wherein thequbit Hamiltonian comprises a constraint by an operator and aneigenvalue of the operator, and wherein the operator commutes with thequbit Hamiltonian; and a plurality of gate operations configured to actupon the plurality of qubits and configured to be implemented by thequantum computer, the plurality of gate operations operable to find thevariational upper bound to the exact energy.

In some embodiments, the operator may comprise a commutation relation ofthe qubit Hamiltonian and a second operator that commutes with the qubitHamiltonian. In some embodiments, the second operator may be a numberoperator. In some embodiments, the second operator may be a total spinoperator. In some embodiments, the operator may be a number operator. Insome embodiments, the operator may be a total spin operator. In someembodiments, the qubit Hamiltonian may comprise a second constraint by asecond operator and a second eigenvalue of the second operator andwherein the second operator commutes with the qubit Hamiltonian.

In an aspect, a system operable to solve a problem is provided. Thesystem may comprise: a quantum computer comprising a plurality ofqubits; a qubit Hamiltonian, wherein one or more coordinates in thequbit Hamiltonian comprises a parametrization in spin coherent states,wherein the parametrization comprises either an operation of one or morequantum logic gates or an expression of the qubit Hamiltonian in Pauli Zrotations, wherein the Hamiltonian is embedded on the quantum computer,wherein one or more eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact energy, and wherein an eigenvalue ofthe qubit Hamiltonian comprises a solution to the problem.

In some embodiments, the spin coherent state comprises an expression inspherical polar coordinates on the Bloch sphere. In some embodiments,the spin coherent state is parameterized by the following:

$ { { { \mspace{20mu}{\text{|}\Omega} \rangle = {{\cos^{2J}( \frac{\theta}{2} )}{\exp\lbrack {{\tan( \frac{\theta}{2} )}e^{i\;\phi}\hat{S}\_} \rbrack}\text{|}{JJ}}} \rangle,{\text{|}\Omega}} \rangle = {\sum\limits_{M = {- J}}^{J}{\begin{pmatrix}{2J} \\{M + J}\end{pmatrix}^{1/2} \times {\cos^{J + M}( \frac{\theta}{2} )}{\sin^{J - M}( \frac{\theta}{2} )}e^{{i{({J - M})}}\phi}\text{|}{JM}}}} \rangle.$

In some embodiments, the operation of one or more quantum logic gatescomprises a qubit mean-field ansatz. In some embodiments, the expressionof the qubit Hamiltonian in Pauli Z rotations comprises an Ising typeHamiltonian. In some embodiments, the expression of the qubitHamiltonian in Pauli Z comprises a quadratic unconstrained boundaryoptimization problem. In some embodiments, the qubit Hamiltonian inPauli Z rotation comprises bias terms and coupling terms for each of theplurality of qubits. In some embodiments, a value of the bias terms anda value of the coupling terms are determined using a classical computer.

In some embodiments, the solution to the problem is a quantum state of amolecule. In some embodiments, the quantum state is a ground state. Insome embodiments, the molecule is an organic optoelectronic material. Insome embodiments, the molecule comprises one or more main groupelements. In some embodiments, the molecule is a polymer.

In some embodiments, the molecule is a molecular crystal. In someembodiments, the qubit Hamiltonian is a mean-field Hamiltonian. In someembodiments, the qubit Hamiltonian comprises a transformation of thefermionic Hamiltonian. In some embodiments, a transformation of thefermionic Hamiltonian comprises using the Jordan-Wigner transformation.In some embodiments, a transformation of the fermionic Hamiltoniancomprises using the Bravyi-Kitaev method. In some embodiments, thequantum computer is a quantum annealer. In some embodiments, theeigenvalue of the reduced Hamiltonian is a global minimum. In someembodiments, the eigenvalue of the reduced Hamiltonian is a localminimum. In some embodiments, the quantum computer is a universal gatequantum computing unit.

In an aspect, a method of solving a problem on a quantum computer isprovided. The method may comprise: providing a qubit Hamiltonian,wherein one or more eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact state energy; parametrizing one ormore coordinates in the qubit Hamiltonian by spin coherent states,wherein parametrizing the one or more coordinates comprises either anoperation of one or more quantum logic gates or expressing the qubitHamiltonian in Pauli Z rotations; directing the qubit Hamiltonian to beembedded on the quantum computer; and receiving an eigenvalue of thequbit Hamiltonian from the quantum computer, wherein the eigenvaluecomprises a solution to the problem.

In some embodiments, the spin coherent state comprises an expression inspherical polar coordinates on the Bloch sphere. In some embodiments,the spin coherent state is parameterized by the following:

$ { { { \mspace{20mu}{\text{|}\Omega} \rangle = {{\cos^{2J}( \frac{\theta}{2} )}{\exp\lbrack {{\tan( \frac{\theta}{2} )}e^{i\;\phi}\hat{S}\_} \rbrack}\text{|}{JJ}}} \rangle,{\text{|}\Omega}} \rangle = {\sum\limits_{M = {- J}}^{J}{\begin{pmatrix}{2\; J} \\{M + J}\end{pmatrix}^{1/2} \times {\cos^{J + M}( \frac{\theta}{2} )}{\sin^{J - M}( \frac{\theta}{2} )}e^{{i{({J - M})}}\phi}\text{|}{JM}}}} \rangle.$

In some embodiments, the operation of one or more quantum logic gatescomprises a qubit mean-field ansatz. In some embodiments, the qubitHamiltonian comprises an Ising type Hamiltonian. In some embodiments,the qubit Hamiltonian comprises a quadratic unconstrained boundaryoptimization problem. In some embodiments, the qubit Hamiltonian inPauli Z rotation comprises bias terms and coupling terms. In someembodiments, the method further comprises determining a value of thebias terms and the coupling terms using a classical computer. In someembodiments, the method further comprises transferring the value of thebias terms and the value of the coupling terms to the quantum computer.

In some embodiments, the solution to the problem is a quantum state of amolecule. In some embodiments, the quantum state is a ground state. Insome embodiments, the molecule is an organic optoelectronic material. Insome embodiments, the molecule comprises one or more main groupelements. In some embodiments, the molecule is a polymer. In someembodiments, the molecule is a molecular crystal.

In some embodiments, the qubit Hamiltonian is a qubit mean-fieldHamiltonian. In some embodiments, the method further comprisestransforming a fermionic Hamiltonian to the qubit Hamiltonian using aJordan-Wigner transformation. In some embodiments, the method furthercomprises transforming a fermionic Hamiltonian to the qubit Hamiltonianusing the Bravyi-Kitaev method. In some embodiments, the method furthercomprises transforming a fermionic Hamiltonian to the qubit Hamiltonianusing the Parity method. In some embodiments, one or more gateoperations are used to determine the lower eigenvalue of the qubitHamiltonian. In some embodiments, a number of gate operations usedscales linearly with a number of fermions in the qubit Hamiltonian. Insome embodiments, the quantum computer is a quantum annealer. In someembodiments, the quantum computer is a universal gate quantum processingunit. In some embodiments, the eigenvalue of the reduced Hamiltonian isa global minimum. In some embodiments, the eigenvalue of the reducedHamiltonian is a local minimum. In some embodiments, the method furthercomprises providing the solution to a user.

In an aspect, a method of solving a problem on a quantum computer isprovided. The method may comprise: providing a qubit Hamiltonian,wherein one or a plurality eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact state energy; providing a set ofentanglers, the entanglers comprising one or a plurality Pauli words;determining a subset of the set of entanglers which reduces a value ofthe one or plurality eigenvalues; truncating the set of entanglers;forming a qubit coupled cluster Hamiltonian comprising the truncated setof entanglers; directing the qubit coupled cluster Hamiltonian to beembedded on the quantum computer; and receiving an eigenvalue of thequbit Hamiltonian from the quantum computer, wherein the eigenvaluecomprises a solution to the problem.

In some embodiments, the method further comprises parameterizing one ormore coordinates of the Hamiltonian by spin coherent states. In someembodiments, the method further comprises expressing the truncated setof entanglers as Pauli z-rotations. In some embodiments, the methodfurther comprises folding an optimization domain in amplitude or phasespace before determining the lower eigenvalue of the qubit coupledcluster Hamiltonian.

In an aspect, a quantum computer is provided. The quantum computer maycomprise: a plurality of qubits; a Hamilton embedded on the quantumcomputer, wherein one or a plurality of eigenvalues of the Hamiltonianis a variational upper bound to an exact energy; and a plurality of gateoperations which may be implemented by the quantum computer, theplurality of gate operations comprising: a set of entanglers comprisingone or a plurality of Pauli Words; and a subset of entanglers, whereinthe subset of entanglers comprises selected Pauli Words which reduce avalue of the one or a plurality of eigenvalues.

In some embodiments, the Hamiltonian comprises a qubit coupled clusterHamiltonian. In some embodiments, the subset of entanglers isparametrized in the Hamiltonian by spin coherent states. In someembodiments, the subset of entanglers is expressed as Pauli z-rotations.In some embodiments, a domain of optimization of the subset ofentanglers is folded in amplitude or phase.

In an aspect, a method of solving a problem on a quantum computer isprovided. The method may comprise: providing a qubit Hamiltonian,wherein one or more eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact state energy; providing an operator,wherein the operator commutes with the qubit Hamiltonian; constrainingthe qubit Hamiltonian with respect to the operator and an eigenvalue ofthe operator; directing the qubit Hamiltonian to be embedded on thequantum computer; and receiving an eigenvalue of the qubit Hamiltonianfrom the quantum computer, wherein the eigenvalue comprises a solutionto the problem.

In some embodiments, the operator comprises a commutation relation ofthe qubit Hamiltonian and a second operator that commutes with the qubitHamiltonian. In some embodiments, the second operator is a numberoperator. In some embodiments, the second operator is a total spinoperator. In some embodiments, the operator is a number operator. Insome embodiments, the operator is a total spin operator. In someembodiments, the method further comprises constraining the qubitHamiltonian with respect to a second operator and a second eigenvalue ofthe second operator and wherein the second operator commutes with thequbit Hamiltonian. In some embodiments, the operator is generated usinga Bravyi-Kitaev or a Jordan-Wigner transformation of a related fermionicoperator. In some embodiments, the method further comprisesparametrizing one or more coordinates in the qubit Hamiltonian by spincoherent states, wherein parametrizing the one or more coordinatescomprises either an operation of one or more quantum logic gates orexpressing the qubit Hamiltonian in Pauli Z rotations. In someembodiments, a domain of optimization of the operator is folded inamplitude or phase.

In an aspect a quantum computer is provided. The quantum computer maycomprise: a plurality of qubits; a qubit Hamilton embedded on thequantum computer, wherein one or a plurality of eigenvalues of the qubitHamiltonian is a variational upper bound to an exact energy, wherein thequbit Hamiltonian comprises a constraint by an operator and aneigenvalue of the operator, and wherein the operator commutes with thequbit Hamiltonian; and a plurality of gate operations configured to actupon the plurality of qubits and configured to be implemented by thequantum computer, the plurality of gate operations operable to find thevariational upper bound to the exact energy.

In some embodiments, the operator comprises a commutation relation ofthe qubit Hamiltonian and a second operator that commutes with the qubitHamiltonian. In some embodiments, the second operator is a numberoperator. In some embodiments, the second operator is a total spinoperator. In some embodiments, the operator is a number operator. Insome embodiments, the operator is a total spin operator. In someembodiments, the qubit Hamiltonian comprises a second constraint by asecond operator and a second eigenvalue of the second operator andwherein the second operator commutes with the qubit Hamiltonian. In someembodiments, the operator is generated using a Bravyi-Kitaev or aJordan-Wigner transformation of a related fermionic operator. In someembodiments, the computer further comprises parametrizing one or morecoordinates in the qubit Hamiltonian by spin coherent states, whereinparametrizing the one or more coordinates comprises either an operationof one or more quantum logic gates or expressing the qubit Hamiltonianin Pauli Z rotations. In some embodiments, a domain of optimization ofthe operator is folded in amplitude or phase.

In an aspect, a method of solving a problem on a quantum computer isprovided. The method may comprise: providing a qubit Hamiltonian,wherein one or a plurality eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact state energy, and wherein the qubitHamiltonian is parameterized by a set of entanglers; determining anfirst entangler of the set of entanglers which reduces a value of theone or plurality eigenvalues; directing the Hamiltonian to be embeddedon the quantum computer, wherein the Hamiltonian comprises the firstentangler; receiving from the quantum computer an amplitude of the firstentangler which produces a first lower eigenvalue of the Hamiltonian;directing the Hamiltonian to be embedded on the quantum computer,wherein the t Hamiltonian comprises the amplitude of the first entanglerand a second entangler of the set of entanglers; receiving from thequantum computer an amplitude of the second entangler which produces asecond lower eigenvalue of the Hamiltonian.

In some embodiments, the method further comprises parameterizing one ormore coordinates of the Hamiltonian by spin coherent states. In someembodiments, the method further comprises folding an optimization domainin amplitude or phase space before determining the lower eigenvalue ofthe Hamiltonian. In some embodiments, the qubit Hamiltonian comprises aqubit coupled cluster Hamiltonian. In some embodiments, the qubitcluster Hamiltonian is transformed as H_(e)+sin(t₁)/2(T₁H_(e)−H_(e)T₁)+(1−cos(t₁))/2 (T₁H₁−H_(e)). In some embodiments, thequantum computer is a quantum annealer. In some embodiments, the quantumcomputer is a universal gate quantum processing unit. In someembodiments, the method further comprises determining a third or morelower eigenvalues of the qubit Hamiltonian using a third or moreentanglers and wherein the first, second, and third lower eigenvaluescomprise a set of lower eigenvalues. In some embodiments, a solution tothe problem is an eigenvalue of the set of lower eigenvalues. In someembodiments, the method further comprises providing the solution to auser. In some embodiments, the solution is a global minimum. In someembodiments, the solution is a local minimum.

In an aspect, a quantum computer is provided. The quantum computer maycomprise: a plurality of qubits; a Hamilton embedded on the quantumcomputer, wherein one or a plurality of eigenvalues of the Hamiltonianis a variational upper bound to an exact energy; and a plurality of gateoperations which may be implemented by the quantum computer, theplurality of gate operations comprising: a set of entanglers whichreduces a value of the one or plurality eigenvalues; an first amplitudeof a first entangler which produces a lower eigenvalue of theHamiltonian; a second entangler of the set of entanglers which reducesthe lower eigenvalue value of the one or plurality eigenvalues of theHamiltonian, wherein the Hamiltonian comprises the first amplitude ofthe first entangler.

In some embodiments, the first and second lower eigenvalues comprise aset of lower eigenvalues. In some embodiments, a solution to the problemis an eigenvalue of the set of lower eigenvalues. In some embodiments,the method further comprises providing the solution to a user. In someembodiments, the solution is a global minimum. In some embodiments, thesolution is a local minimum. In some embodiments, one or morecoordinates of the Hamiltonian is parameterized by spin coherent states.

In an aspect, a system operable to solve a problem is provided. Thesystem may comprise: a computer comprising: a qubit Hamiltonian, whereinone or more coordinates in the qubit Hamiltonian comprises aparametrization in spin coherent states, one or more eigenvalues,wherein the one or more eigenvalues of the qubit Hamiltonian is avariational bound to an exact energy, and a solution to the problem,wherein the solution comprises an eigenvalue of the qubit Hamiltonian.

In some embodiments, the computer is classical computer. In someembodiments, the computer simulates the operation of the quantumcomputer of any aspect or embodiment disclosed herein. In someembodiments, the system further comprises, the quantum computer of anyaspect or embodiment disclosed herein. In some embodiments, theclassical computer is operable to control the quantum computer. In someembodiments, the solution is provided to a user.

In some embodiments, the parameterization in spin coherent statescomprises an expression in spherical polar coordinates on the Blochsphere. In some embodiments, the spin coherent states are parameterizedby the following:

$ { { { \mspace{20mu}{\text{|}\Omega} \rangle = {{\cos^{2J}( \frac{\theta}{2} )}{\exp\lbrack {{\tan( \frac{\theta}{2} )}e^{i\;\phi}\hat{S}\_} \rbrack}\text{|}{JJ}}} \rangle,{\text{|}\Omega}} \rangle = {\sum\limits_{M = {- J}}^{J}{\begin{pmatrix}{2\; J} \\{M + J}\end{pmatrix}^{1/2} \times {\cos^{J + M}( \frac{\theta}{2} )}{\sin^{J - M}( \frac{\theta}{2} )}e^{{i{({J - M})}}\phi}\text{|}{JM}}}} \rangle.$

In some embodiments, the solution is a local minimum. In someembodiments, the solution is a global minimum. In some embodiments, thequbit Hamiltonian is parameterized in Pauli Z rotations. In someembodiments, the qubit Hamiltonian comprises a QUBO, k-local Isingmodel, or HOBO model. In some embodiments, the qubit Hamiltonianparametrized in Pauli Z rotation comprises bias terms and couplingterms. In some embodiments, the computer comprises a value of the biasterms and the coupling terms. In some embodiments, the value of the biasterms and the value of the coupling terms are transferred to the quantumcomputer. In some embodiments, the Hamiltonian comprises a qubit coupledcluster Hamiltonian. In some embodiments, the qubit cluster Hamiltonianis transformed as H_(e)+sin(t₁)/2 (T₁H_(e)−H_(e)T₁)+(1−cos(t₁))/2(T₁H₁−H_(e)). In some embodiments, the qubit coupled cluster Hamiltoniancomprises a set of entanglers parametrized by spin coherent states. Insome embodiments, the set of entanglers comprises one or a plurality ofPauli Words. In some embodiments, the set of entanglers comprises asubset of entanglers, wherein the subset of entanglers comprisesselected Pauli Words which reduce a value of the one or a plurality ofeigenvalues. In some embodiments, the subset of entanglers is expressedas Pauli z-rotations. In some embodiments, a domain of optimization ofthe qubit Hamiltonian is folded in amplitude or phase.

In an aspect, a computer-implemented method of solving a problem isprovided. The method may comprise: providing a qubit Hamiltonian,wherein one or more eigenvalues of the qubit Hamiltonian is avariational upper bound to an exact state energy; parametrizing one ormore coordinates in the qubit Hamiltonian by spin coherent states; andproviding a solution, wherein the solution to the problem comprises aneigenvalue of the qubit Hamiltonian.

In some embodiments, the solution to the problem is provided to a user.In some embodiments, the method further comprises directing the qubitHamiltonian to be embedded on the quantum computer of aspect orembodiment disclosed herein. In some embodiments, the method furthercomprises receiving the solution from the quantum computer of any aspector embodiment disclosed herein. In some embodiments, the method furthercomprises simulating the quantum computer of any aspect or embodimentdisclosed herein.

In some embodiments, the parameterizing comprises an expression inspherical polar coordinates on the Bloch sphere. In some embodiments,the parameterizing in spin coherent states comprises the following:

$ { { { \mspace{20mu}{\text{|}\Omega} \rangle = {{\cos^{2J}( \frac{\theta}{2} )}{\exp\lbrack {{\tan( \frac{\theta}{2} )}e^{i\;\phi}\hat{S}\_} \rbrack}\text{|}{JJ}}} \rangle,{\text{|}\Omega}} \rangle = {\sum\limits_{M = {- J}}^{J}{\begin{pmatrix}{2\; J} \\{M + J}\end{pmatrix}^{1/2} \times {\cos^{J + M}( \frac{\theta}{2} )}{\sin^{J - M}( \frac{\theta}{2} )}e^{{i{({J - M})}}\phi}\text{|}{JM}}}} \rangle.$

In some embodiments, the solution is a local minimum. In someembodiments, the solution is a global minimum. In some embodiments, theparametrizing the one or more coordinates comprises expressing the qubitHamiltonian in Pauli Z rotations. In some embodiments, the parametrizingcomprises a QUBO, k-local Ising model, or HOBO model. In someembodiments, the parametrizing comprises either an operation of one ormore quantum logic gates. In some embodiments, the Hamiltonian comprisesa qubit coupled cluster Hamiltonian. In some embodiments, the methodfurther comprises transforming the qubit cluster Hamiltonian asH_(e)+sin(t₁)/2 (T₁H_(e)−H_(e)T₁)+(1−cos(t₁))/2 (T₁H₁−H_(e)). In someembodiments, the qubit coupled cluster Hamiltonian comprises a set ofentanglers parametrized by spin coherent states. In some embodiments,the set of entanglers comprises one or a plurality of Pauli Words. Insome embodiments, the method further comprises selecting Pauli Wordswhich reduce a value of the one or a plurality of eigenvalues to form asubset of entanglers. In some embodiments, the subset of entanglers isexpressed as Pauli z-rotations. In some embodiments, the method furthercomprises folding a domain of optimization of the qubit Hamiltonian inamplitude or phase.

In another aspect, a non-transitory computer readable storage mediumwith instructions stored thereon is provided. In some embodiments, theinstructions when executed by a quantum computer may be configured toperform the methods above. In some embodiments, the instructions whenexecuted by a classical computer may be configured to perform themethods above on a quantum computer. In some embodiments, thenon-transitory computer readable storage medium may comprise a classicalstorage unit. In some embodiments, the non-transitory computer readablestorage medium may be quantum storage unit. In some embodiments, theinstructions when executed may be configured to provide the solution toa user.

INCORPORATION BY REFERENCE

All publications, patents, and patent applications mentioned in thisspecification are herein incorporated by reference to the same extent asif each individual publication, patent, or patent application wasspecifically and individually indicated to be incorporated by reference.To the extent publications and patents or patent applicationsincorporated by reference contradict the disclosure contained in thespecification, the specification is intended to supersede and/or takeprecedence over any such contradictory material.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features of the invention are set forth with particularity inthe appended claims. A better understanding of the features andadvantages of the present invention will be obtained by reference to thefollowing detailed description that sets forth illustrative embodiments,in which the principles of the invention are utilized, and theaccompanying drawings of which:

FIG. 1 shows an example method for solving a problem.

FIG. 2 shows an implementation of the method of FIG. 1.

FIG. 3 shows an example method for implementing an Ising model on aquantum annealer.

FIG. 4 shows an example method for implementing a QUBO model on aquantum annealer.

FIG. 5 shows an example implementation of a method of solving a problemusing the quantum mean-field ansatz.

FIG. 6 shows a second example implementation of a method of solving aproblem using the quantum mean-field (QMF) ansatz.

FIG. 7 shows the two lowest eigenstates of the Hamiltonian for H₂, theminimum of the corresponding QMF functional, and experimental data froma Rigetti quantum computer.

FIG. 8 shows the two lowest eigenstates of the Hamiltonian for H₂, theminimum of the corresponding QMF functional, and experimental data froma Rigetti quantum computer, where the QMF functional has beenconstrained by S²=0.

FIG. 9 shows the two lowest eigenstates of the Hamiltonian for H₂ ⁺, theminimum of the corresponding QMF functional, and experimental data froma Rigetti quantum computer, where the QMF functional has beenconstrained by N=1.

FIG. 10 shows a classical computer system coupled to a quantum computersystem that is programmed or otherwise configured to implement methodsprovided herein.

FIG. 11 shows correlated mean-field energy E_(cQMF) ^(x) ¹ ^(x) ⁰ (τ)for the generator {circumflex over (T)}₁=x₁z₀ for H₂ in a minimal basis.

FIG. 12 shows correlated mean-field energy E_(cQMF) ^(x) ¹ ^(x) ⁰ (τ)for the generator {circumflex over (T)}₁=x₁z₀ for H₂ in a minimal basis.

FIG. 13A and FIG. 13B show an example entanglement ansatz based onalpha-beta molecular orbital blending.

FIG. 14 shows an example method comprising ranking the Pauli operatorsby their contribution to the qubit coupled-cluster (QCC) Hamiltonian.

FIG. 15 shows a calculated potential energy curve for the bondstretching coordinate of the H2 molecule using the example QCCentanglement ansatz of FIG. 14. FIG. 15 also shows results for the QMFmethod, spin constrained QMF method, and the exact potential energysurface

FIG. 16 shows an example method comprising ranking the Pauli operatorsby their contribution to the QCC Hamiltonian and implementing the QCCmethod on a quantum annealer.

FIG. 17 shows a calculated potential energy curve for the bondstretching coordinate of the LiH molecule using the example QCCentanglement ansatz of FIG. 14. FIG. 17 also shows results for the QMFmethod, spin constrained QMF method, and the exact potential energysurface

FIG. 18 shows an example method comprising ranking the Pauli operatorsby their contribution to the QCC Hamiltonian, folding the optimizationdomain, and implementing the QCC method on a quantum annealer.

FIG. 19 shows an example implementation of a constraint usingprojectors.

FIG. 20 shows the results of implementing the QCC method without domainfolding (2010)—solved on a classical computer—and with domain folding(2020) solving the discrete optimization on a quantum annealer of LiHfor a bond distance of 1.45 Å, and each condition was repeated 101times.

DETAILED DESCRIPTION

While various embodiments of the invention have been shown and describedherein, it will be obvious to those skilled in the art that suchembodiments are provided by way of example only. Numerous variations,changes, and substitutions may occur to those skilled in the art withoutdeparting from the invention. It should be understood that variousalternatives to the embodiments of the invention described herein may beemployed.

Unless otherwise defined, all technical terms used herein have the samemeaning as commonly understood by one of ordinary skill in the art towhich this invention belongs. As used in this specification and theappended claims, the singular forms “a,” “an,” and “the” include pluralreferences unless the context clearly dictates otherwise. Any referenceto “or” herein is intended to encompass “and/or” unless otherwisestated.

Whenever the term “at least,” “greater than,” or “greater than or equalto” precedes the first numerical value in a series of two or morenumerical values, the term “at least,” “greater than” or “greater thanor equal to” applies to each of the numerical values in that series ofnumerical values. For example, greater than or equal to 1, 2, or 3 isequivalent to greater than or equal to 1, greater than or equal to 2, orgreater than or equal to 3.

Whenever the term “no more than,” “less than,” or “less than or equalto” precedes the first numerical value in a series of two or morenumerical values, the term “no more than,” “less than,” or “less than orequal to” applies to each of the numerical values in that series ofnumerical values. For example, less than or equal to 3, 2, or 1 isequivalent to less than or equal to 3, less than or equal to 2, or lessthan or equal to 1.

Incorporation by reference is expressly limited to the technical aspectsof the materials, systems, and methods described in the mentionedpublications, patents, and patent applications and does not extend toany lexicographical definitions from the publications, patents, andpatent applications. Any lexicographical definition appearing in thepublications, patents, and patent applications that is not alsoexpressly repeated in the instant disclosure should not be treated assuch and should not be read as defining any terms appearing in theaccompanying claims.

In the following detailed description, reference is made to theaccompanying figures, which form a part hereof. In the figures, similarsymbols typically identify similar components, unless context dictatesotherwise. The illustrative embodiments described in the detaileddescription, figures, and claims are not meant to be limiting. Otherembodiments may be utilized, and other changes may be made, withoutdeparting from the scope of the subject matter presented herein. It willbe readily understood that the aspects of the present disclosure, asgenerally described herein, and illustrated in the figures, can bearranged, substituted, combined, separated, and designed in a widevariety of different configurations, all of which are explicitlycontemplated herein.

Classical computers represent information in binary variables as 0 and1, called bits, and perform gate operations upon these bits in certaincombinations to represent numerical data. A qubit as described herein isalso a two state system; however, rather than a conventional bit whichis either zero or one, a qubit may comprise a quantum mechanicalsuperposition of both 0 and 1. The superposition potentially allows moreinformation to be encoded than a classical bit. Additionally, a quantumcomputer may compute transformations (e.g. gate operations) on thesuperposition, which may result in significant increases in processingspeed as a result of inherent parallelization.

This inherent parallelization can be used to reduce the computation costof some computations. In an example, computer simulations can be madeof, for example, protein folding, quantum systems, ecological habitats,economic processes, etc. In particular aspects, quantum computers mayreduce the computational cost of calculations concerning the electronicstructure of molecules. Chemical electronic structure calculations solvefor the eigenstates of an interacting, many-body fermionic Hamiltonian.These problems are challenging in several aspects. Importantly,computational cost of quantum simulations using classical computers andconventional numerical methods suffer from poor scaling with system size(exact solutions scale exponentially and approximate solutions may scalepoorly and may provide insufficient accuracy); however, a quantumcomputer may be able to simulate quantum systems with relatively highaccuracy and in polynomial time.

The present disclosure provides methods and systems to transform thefermionic Hamiltonian and embed it into quantum computer in such a waythat the number of qubits used to solve a problem is reduced. In anexample, the problem may comprise an electronic structure problem. Theproblem may comprise a quantum chemical problem. The problem maycomprise a quantum mechanical simulation of a molecular system. Solutionto such problems may comprise the structure and/or properties of matter.Example properties include but are not limited to absolute and relativeenergies, ground states, excited states, ionization potentials, chargedensity distributions, dipoles, higher-multipole moments, polarizabilitytensors, HOMO-LUMO gaps, ionization potentials, isoelectronic surfaces,infrared spectra, ultraviolet-visible (UV-Vis) spectra, spectra in anyother energy range of the electromagnetic spectrum, vibrationalfrequencies, reactivities, collision cross sections, thermodynamicproperties, potential energy surfaces, bond energies, bond lengths, bondangles, rates of reaction, and time evolution of the precedingproperties including for example: charge transport, adiabatic chemicaldynamics, vibronic coupling, folded space optimization, and variouscalculation steps thereof.

Although specific examples of fermionic Hamiltonians are describedherein, the methods and systems disclosed herein may be applicable tothe solution of any fermionic Hamiltonian. A fermionic Hamiltonian canbe formulated to solve a problem of any of the following non-limitingexamples: in the field of quantum chemistry, for drug discovery, foroptoelectronic materials discovery, to determine an optimum emittermaterial based on the HOMO-LUMO gap, for economic modeling, for proteinfolding applications, etc.

The methods and systems disclosed herein are applicable to Hartree-Fockcalculations; however, the methods and systems disclosed herein can alsobe applied to post-Hartree-Fock calculations, to density functionaltheory (DFT) calculations, to time-dependent Hartree-Fock calculations,to time dependent density functional theory (TD-DFT) calculations, or toother types of quantum chemical calculation.

The methods disclosed herein can in principal be performed on anycomputer, for example those disclosed herein, which is capable ofperforming or simulating the performance of the qubit operations used toperform that method. Systems and methods disclosed herein with respectto solution of an Ising or quadratic unconstrained boundary optimization(QUBO) type problem may be performed on a quantum annealer or a computeroperable to simulate a quantum annealer (e.g. a simulated annealer).Systems and methods disclosed herein with respect to solution of aproblem using a qubit mean-field ansatz can be performed or adapted tobe performed on any quantum computer or a computer operable to simulatethe qubit operations (e.g. gate operations, etc.) of a quantum computer.

A “qubit” as defined herein is a unit of quantum information encoded ina physical system. Information may be encoded by any physical systemcapable of the quantum effects of superposition between quantum statesand entanglement with one or more qubits. A qubit has a probability ofbeing in a state 0 and a probability of being in a state 1. A state of aqubit can be represented as a linear combination of states 0 and 1scaled by coefficients, where the square of the coefficient is theprobability of measuring the corresponding state. Additionally oralternatively, the state of a qubit can be viewed as a vector on theBloch sphere where the north and south poles of the sphere comprise thestates 0 and 1. One can view changes in the relative probability of eachstate as rotations of the vector on the Bloch sphere. Physicalimplementations of qubits include superconducting charge qubits,superconducting flux qubits, superconducting phase qubits, nuclear spinstates, atomic spins states, electron spin states, electron numberstates, squeezed states of light, polarization encoded photons, quantumdot spin states, etc.

A “quantum logic gate” as defined herein is any device or processeswhich can perform a logic operation on a qubit as part of a quantumcircuit. Typically quantum logic gates can be represented as unitarymatrices. Logic operations on the set of qubits typically includesuperposition gates and entanglement gates. Superposition gates act upona qubit to achieve a superposition and/or change the relativeprobability of each of the two pole states. Superposition gates includePauli Operators (X, Y, Z), Hadamard gates, and Pauli rotation gates,etc. Entanglement gates couple two or more qubits. Examples ofentanglement gates comprise for example cZ gates, CNOT gates, etc.Varying types of quantum computers may be capable of all or a subset thegates listed herein. Other types of gates not listed herein may bepossible.

A “Pauli word” as defined herein is any product of one or more Paulioperators of different qubits. In some examples, a Pauli word{circumflex over (P)}_(I) may be represented by the following relation:

{circumflex over (P)} ₁= . . . {circumflex over (ω)}_(P)^((I)){circumflex over (ω)}_(R) ^((I))

The term {circumflex over (ω)}_(R) ^((I)) may comprise one of the x, y,z Pauli operators for the r^(th) qubit. In some cases, the Pauli wordmay be a generator. The generator may be a generator of entanglement.The entanglement operator may take the form of a Unitary operatorparameterized as follows: Û_(ENT)=e^(−iτ{circumflex over (P)}) ¹ . Insome cases, the Pauli word comprises a product of two Pauli operatorsfor two qubits. In some cases, the Pauli Word may be of the formx_(P)x_(R), x_(P)z_(R), z_(p)z_(R), x_(p)y_(R), y_(P)z_(R), y_(P)y_(R),z_(P)z_(R), etc. where P and R are qubit indices and P does not equal R.In some cases, the Pauli word comprises a product of three Paulioperators. In some cases, the Pauli word comprises a product of fourPauli operators. The Pauli word may be a product of less than 5 Paulioperators, less than 10 Pauli operators, or less than 100 Paulioperators. The amplitude τ may be related to the microwave pulseduration. In some cases, it may be the time evolution of the unitaryoperator, which may fictitious in some cases. It may be the amplitude ofa rotation gate within the entanglement operator. Every entanglementoperator may comprise a different amplitude.

A “quantum computer” or “quantum processing unit (QPU)” as definedherein is any device that harnesses one or more quantum effects (e.g.superposition, entanglement, etc.) to perform a computation. A quantumcomputer as used herein may be any non-classical computer. Anon-classical computer may be capable of using one or more quantumeffects to perform a computation but may not be a universal quantumcomputer, as disclosed herein below. A quantum computer may comprise oneor a plurality of qubits. For instance, a quantum computer may compriseat least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70,80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, or morequbits. A quantum computer may comprise at most about 1,000, 900, 800,700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10,9, 8, 7, 6, 5, 4, 3, 2, or 1 qubits. A quantum computer may comprise anumber of qubits that is within a range defined by any two of thepreceding values. In some embodiments, a quantum computer may comprise aquantum processing unit comprising one or a plurality of qubits. In theso-called “circuit model” of quantum computing, a quantum computer canbe viewed as performing a set of logic operations on a set of qubits forthe purpose of solving a problem. A quantum computer may compriseoperations for manipulating the superposition and/or entanglement of thequbits. Such operations may comprise electrical pulses, photons, etc.Additionally or alternatively, the potential energy of the qubit may becontrollably varied. The potential energy may comprise biases (one-qubitterms) and coupling terms (two or more qubit terms). Varying forms ofquantum computers have been proposed and realized, among which includeuniversal quantum computers and quantum annealers. Example quantumcomputers include IBM's Q, D-Wave's Q2000, Fujitsu's Digital Annealer,19Q-Acorn by Rigetti, Xanadu's quantum photonic processors, etc.

In some embodiments, a quantum computer may comprise one or moreadiabatic quantum computers, quantum gate arrays, one-way quantumcomputers, topological quantum computers, quantum Turing machines,superconductor-based quantum computers, trapped ion quantum computers,trapped atom quantum computers, optical lattices, quantum dot computers,spin-based quantum computers, spatial-based quantum computers,Loss-DiVincenzo quantum computers, nuclear magnetic resonance (NMR)based quantum computers, solution-state NMR quantum computers,solid-state NMR quantum computers, solid-state NMR Kane quantumcomputers, electrons-on-helium quantum computers,cavity-quantum-electrodynamics based quantum computers, molecular magnetquantum computers, fullerene-based quantum computers, linear opticalquantum computers, diamond-based quantum computers, nitrogen vacancy(NV) diamond-based quantum computers, Bose-Einstein condensate-basedquantum computers, transistor-based quantum computers, andrare-earth-metal-ion-doped inorganic crystal based quantum computers. Aquantum computer may comprise one or more of: quantum annealers, Isingsolvers, optical parametric oscillators (OPO), and gate models ofquantum computing.

A “universal quantum computer” as defined herein is a quantum computerthat can simulate the operation of any other quantum computer. In somecases, a quantum computer, which may not be a universal quantumcomputer, may be capable of simulating a gate operation which it may notbe capable of performing directly. Methods disclosed herein may beperformed by any quantum computer which includes or is capable ofsimulating a specified operation. Methods disclosed herein may, bydefinition, be performed by a universal quantum computer.

A “quantum annealer” as defined herein is a computer operable to solve aproblem in the form of a binary optimization problem. In some cases, thequantum annealer may be limited to X and Z gates. In some cases, toperform the annealing process, X operators may be used to push qubitsinto a superposition, and then, in some cases, may not be used again onthe qubits. Quantum annealers may be limited to solving binaryoptimization problems in the form, for example, of an Ising model or aQUBO problem. However, despite potential limitations, quantum annealersmay be more accurate and faster with current hardware. In some cases,methods described herein may be performed on a quantum annealer.

In some cases, methods described herein may be performed on a classicalcomputer using quantum inspired algorithms. For example, methodsdescribed herein may be performed on a simulated annealer. In somecases, a classical computer (e.g. CPU, GPU, FPGA, Asyc, etc.) that runsa particular process (simulated annealing, parallel tempering, simulatedquantum annealing) may also be capable of solving the ground stateconfiguration of the Ising, QUBO or high order binary optimization(HOBO) problem. These quantum inspired algorithms may receive the sameor similar Ising, QUBO or HOBO model and may solve the ground stateusing various methods. In some cases, these methods may comprisesimulated annealing. Simulated annealing uses classical thermalfluctuations to guide the qubit Hamiltonian represented in Pauli Zrotations to the ground state energy. In some cases, simulated annealingmay be performed without using quantum effects. These methods may or maynot be more efficient than methods used in quantum annealers.

Methods and Systems for Solving a Problem:

Described herein in certain embodiments is a system operable to solve aproblem. The system may comprise a quantum computer comprising aplurality of qubits, a qubit Hamiltonian, wherein one or morecoordinates in the qubit Hamiltonian comprises a parametrization in spincoherent states, wherein the parametrization comprises either anoperation of one or more quantum logic gates or an expression of thequbit Hamiltonian in Pauli Z rotations; wherein the Hamiltonian isembedded on the quantum computer; wherein one or more eigenvalues of thequbit Hamiltonian is a variational upper bound to an exact energy; andwherein a lower eigenvalue of the qubit Hamiltonian comprises a solutionto the problem.

Also described herein in certain embodiments is a method of solving aproblem on a quantum computer. The method may comprise: providing aqubit Hamiltonian, wherein one or more eigenvalues of the qubitHamiltonian is a variational upper bound to an exact state energy;parametrizing one or more coordinates in the Hamiltonian by spincoherent states, wherein parametrizing the one or more coordinatescomprises either an operation of one or more quantum logic gates orexpressing the qubit Hamiltonian in Pauli Z rotations; embedding thequbit Hamiltonian on the quantum computer; and determining a lowereigenvalue of the qubit Hamiltonian, wherein the lower eigenvaluecomprises a solution to the problem.

Also, described herein in certain embodiments is a system operable tosolve an electronic structure problem, the system comprising: a quantumcomputer comprising a plurality of qubits, wherein a given qubit of theplurality of qubits comprises a bias and a coupling between the givenqubit and another qubit; wherein the bias and coupling of the givenqubit of the plurality of qubits comprise parameters in a reducedHamiltonian; wherein the reduced Hamiltonian comprises a transformationof a qubit mean-field Hamiltonian, and wherein one or more eigenvaluesof the qubit mean-field Hamiltonian is a variational upper bound to anexact energy; and wherein a lower eigenvalue of the reduced Hamiltoniancomprises a solution to the electronic structure problem.

Also, described herein in certain embodiments is a method of solving aproblem on a quantum computer, the method comprising: providing thequantum computer, wherein the quantum computer comprises a plurality ofqubits, and wherein a given qubit of the plurality of qubits comprises abias and a coupling between the given qubit and another qubit; providinga qubit mean-field Hamiltonian; transforming the qubit mean-fieldHamiltonian to generate a reduced Hamiltonian, wherein one or moreeigenvalues of the qubit mean-field Hamiltonian is a variational upperbound to an exact energy; embedding the reduced Hamiltonian on thequantum computer, wherein the bias and coupling of the given qubit ofthe plurality of qubits comprise parameters in the reduced Hamiltonian;and determining a lower eigenvalue of the reduced Hamiltonian, whereinthe lower eigenvalue comprises a solution to the electronic structureproblem.

Also described herein in certain embodiments is a method of solving aproblem on a quantum computer, the method comprising: providing a qubitmean-field Hamiltonian, wherein one or more eigenvalues of the qubitmean-field Hamiltonian is a variational upper bound to an exact energy;providing a qubit mean-field ansatz, wherein the ansatz is parameterizedby one or more spin coherent states; embedding the qubit mean-fieldHamiltonian on the quantum computer; and using the qubit mean-fieldansatz to determine a lower eigenvalue of the qubit mean-fieldHamiltonian, wherein the lower eigenvalue comprises a solution to theproblem.

Example matter may include atoms, molecules, and groups of atoms and/ormolecules. Molecules may comprise main group elements, transition metalelements, post-transition metal elements, etc. Example matter includesextended states of matter including condensed matter. Molecules maycomprise polymers, molecular crystals, organometallic compounds, smallmolecules, organic compounds, materials for organic photovoltaics,optoelectronic materials, etc. Optoelectronic materials may be used inthe manufacture of light emitting diodes, for example organic lightemitting diodes. Examples of such materials include organic materials,such as small molecule organic materials and organic polymers. Examplesof suitable organic materials include polycyclic aromatic compoundsincluding organic molecules which may optionally include one or moreheteroatoms, such as nitrogen (N), sulfur (S), oxygen (O), phosphorus(P), fluorine (F), and aluminum (Al). Examples of organometalliccompounds include organometallic complexes or metal coordinationcomplexes. Examples of such complexes include those formed by a metalliccoordination center and ligands surrounding the coordination center.Examples of an atom or ion which may form the coordination centerinclude, but are not limited to, iridium (Ir), Zn, rhodium (Rh), Al,beryllium (Be), rhenium (Re), ruthenium (Ru), boron (B), P, Cu, osmium(Os), gold (Au), and platinum (Pt). In complexes or metal coordinationcomplexes, a dative bond may be formed between the coordination centerand one or more atoms of the surrounding ligands. Examples of bondswhich may be formed between the coordination center and one or moreatoms of the surrounding ligands include, but are not limited to, thoseformed between a metallic atom of the coordination center and carbon,nitrogen, or oxygen. Specifically, examples of such bonds include thoseformed between Al and O, Al and N, Zn and O, Zn and N, Zn and C, Be andO, Be and N, Ir and N, Ir and C, Ir and O, Cu and N, B and C, Pt and N,Pt and O, Os and N, Ru and N, Re and N, Re and O, Re and C, Cu and P, Auand N, Os and C, etc. Other example bonds may include those of organiccompounds, such as C and C, C and N, C and O, C and H, C and P, O and H,N and H, O and N, etc. Example molecules include fullerenes. Fullerenesmay comprise C60, C70, C76, C84, single-wall carbon nanotubes,multi-wall carbon nanotubes, and any combination thereof. Optoelectronicmaterials may comprise quantum dots.

Shown in FIG. 1, a method 100 of solving a problem may include providinga fermionic Hamiltonian 120, transformation of the fermionic Hamiltonianto qubit operators 140, transformation of the fermionic Hamiltonian inqubit operators to a mean-field Hamiltonian 160, and embedding themean-field Hamiltonian onto a quantum computer 180. Such a method mayimprove upon existing methods for solving electronic structure problemson a computer by adapting the problem to available hardware, reducingcomputational cost, and reducing the number of qubits used to solveelectronic structure problems for larger number of atoms.

Fermionic Hamiltonian:

Shown in FIG. 2 is example implementation 200 of the method of FIG. 1.Operation 120 in a method 100 may comprise operations 210 and 220 of themethod 200. Operation 210 may comprise expressing the one and twoelectron integrals in a basis which may be convenient for computation.In some cases, the computation basis may be derived from Fock-orbitals,molecular orbitals, or atomic orbitals. Basis functions may be providedfrom any of for example, Gaussian-type orbitals, Slater-type orbitals,numerical atomic orbitals, etc. The basis set may comprise a minimalbasis set such as for example, STO-3G, STO-4G, and STO-nG, where n is aninteger. The integer n may comprise for example, an integer of at leastabout 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90,100, or more, an integer of at most about 100, 90, 80, 70, 60, 50, 40,30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1, or an integer that is within arange defined by any two of the preceding values. The Slater-type basissets may comprise polarized versions. The basis set may comprise a splitvalence basis set such as a Pople basis set, for example, 3−21G, 3−21G*,3−21G**, 3−21+G, 3−21++G, 3−21+G*, 3−21+G**, 4−21G, 4−31G, 6−21G, 6−31G,6−31G*, 6−31+G*, 6−31G(3df, 3pd), 6−311G, 6−311G*, 6−311+G*, etc. Thebasis set may comprise a Dunning type basis set for example, cc-pVDZ,cc-pVTZ, cc-pVQZ, cc-pV5Z, aug-cc-pVDZ, cc-pCVDZ, etc.

Operation 220 may comprise providing the fermionic Hamiltonian. Afermionic Hamiltonian may be derived from multiple sources. One examplemay be the second order, fermionic Hamiltonian, shown below.

$\hat{H} = {{\sum\limits_{i,j}{h_{i,j}a_{i}^{\dagger}a_{j}}} + {\frac{1}{2}{\sum\limits_{i,j,k,l}{h_{i,j,k,l}a_{i}^{\dagger}a_{j}^{\dagger}a_{k}a_{l}}}}}$

However, in some cases, higher or lower order Hamiltonians may beprovided. The terms and h_(i,j,k,l) may be defined as follows:

${h_{i,j} = {\int{{\psi_{i}^{*}(x)}\hat{h}{\psi_{i}(x)}{dx}}}},{h_{i,j,k,l} = {\int{{\psi_{i}^{*}( x_{1} )}{\psi_{j}^{*}( x_{2} )}\frac{1}{r_{1,2}}{\psi_{k}( x_{1} )}{\psi_{l}( x_{1} )}{dx}_{1}{dx}_{2}}}}$

where ĥ is the one-electron Hamiltonian

$\hat{h} = {{- \frac{1}{2}}{\nabla_{r}^{2}{- {\sum\limits_{\alpha}{\frac{z_{\alpha}}{{r - R_{\alpha}}}.}}}}}$

In some cases, the one and two electron integrals may be computed on aclassical computer and pushed to the quantum computer. In other cases,the one and two electron integrals are calculated on the quantumcomputer.

A classical computer of the present disclosure is described further withreference to the section “classical control systems”.

Transformation of the Fermionic Hamiltonian to a Qubit Representation:

At operation 230 of method 200 the fermionic Hamiltonian may beexpressed in qubit operators rather than in a fermionic representationby with a basis transformation. Operation 230 can include an embodiment,variation, or example of operation 160 of the method 100. After such atransformation, finding the eigenstates of the qubit Hamiltonian may betantamount to solving the quantum chemistry problem. At least twoexample methods for transforming the Hamiltonian are provided herein,the Jordan-Wigner transformation and the Bravyi-Kitaev method; however,additional methods of transforming the Hamiltonian may be possible. Theresulting qubit Hamiltonian may be a linear equation comprising Paulioperators of X, Y, and Z and may maintain the size of the Hilbert spaceof 2^(k), where k is a number of single-particle states. In some cases,the Hamiltonian is of second quantization form as shown above.

In a first example, the creation and annihilation operators may bemapped to operations on the qubits as Pauli rotation operators, below:

{circumflex over (Q)}+|1

0|=½(σ^(x) −iσ ^(y)); {circumflex over (Q)}+|1

1|=½(σ^(x) −iσ ^(y))

In the occupation number basis, each qubit may store the occupationnumber of an orbital indexed by the qubit; however, other bases arepossible. This mapping of interacting fermions to spin operatorscomprises the Jordan-Wigner transformation. In some cases, theJordan-Wigner representation may result in non-locality of the parityoperator.

In a second example, the creation and annihilation operators may betransformed by encoding fermionic states in a parity basis rather thanan occupation number basis. However, the scaling of the number of qubitoperations to simulate a single fermionic operation may be the same orsimilar in both the parity basis and the occupation number basis (e.g.scale with O(k), where k is the number of single particle states). Insome cases, the parity basis may result in non-locality of theoccupation number operator.

In a third example, qubits may store partial sums of occupation numberand parity. In one such example, the Bravyi-Kitaev basis may reduce thenumber of qubit operations used to simulate a single fermionic operatorto O(log k), as detailed in Seeley (J. T. Seeley, M. J. Richard, and P.J. Love, J. Chem. Phys. 137, 224109 (2012)), which is hereinincorporated by reference in its entirety. Such a method may combineboth aspects of the parity basis and the number basis, such that boththe parity and the number operator may be evaluated on a reduced set ofqubits which may be updated for any single operation. The partial summay be evaluated up to an index containing a full parameterization ofthe number and the parity operator of the qubit operated on.

Operation 160 of the method 100 may comprise operation 240 of the method200. At an operation 240 of the method 200, the mean-field Hamiltonianmay be transformed to the qubit representation by a method similar tothose disclosed herein with respect to operation 230. By any methoddisclosed herein, the qubit Hamiltonian Ĥ assumes the general form:

$\hat{H} = {\sum\limits_{I}{C_{I}{\hat{T}}_{I}}}$

where molecular-integrals may depend on coefficients C_(I) and operatorsT_(I) which may be products of qubit spin operators {circumflex over(T)}=ω₁ . . . ω_(m), 0≤m≤n, where n is the number of qubits.Coefficients C_(I) may be spatially dependent. The qubit Hamiltonianderived via methods described herein may be isospectral to the fermionicHamiltonian, such that every eigenvalue of the qubit Hamiltonian is aneigenvalue of the fermionic Hamiltonian. The qubit Hamiltonian maycomprise a spin orbital Hamiltonian.

Spin Coherent States

A spin coherent state, also called a “Bloch state” or a “Bloch CoherentState”, for a single particle with spin J (J≥0 is integer ofhalf-integer) is defined by the action of an appropriately scaledexponent of the lowering operator Ŝ_ onto the normalized eigenfuction ofŜ_(z) operator, Ŝ_(z)|JM

=M|JM

, with maximal projection M=J:

$ { { { {\text{}\Omega} \rangle = {{\cos^{2J}( \frac{\theta}{2} )}{\exp\lbrack {{\tan( \frac{\theta}{2} )}e^{i\;\phi}\hat{S}\_} \rbrack}\text{|}{JJ}}} \rangle = {\text{|}\Omega}} \rangle = {\sum\limits_{M = {- J}}^{J}{\begin{pmatrix}{2\; J} \\{M + J}\end{pmatrix}^{1/2} \times {\cos^{J + M}( \frac{\theta}{2} )}{\sin^{J - M}( \frac{\theta}{2} )}e^{{i{({J - M})}}\phi}\text{|}{JM}}}} \rangle.$

|JM

normalized as:

$ { {|{JM}} \rangle = {{\begin{pmatrix}{2\; J} \\{M + J}\end{pmatrix}^{1/2}\lbrack {( {J - M} )!} \rbrack}{\hat{S}}^{\underset{\_}{J} - M}\text{|}{JJ}}} \rangle$

The above constitutes an (over)complete non-orthogonal set of states ona unit sphere parameterized by spherical polar angles, Ω=(ϕ, θ), 0≤ϕ≤2π,0≤θ≤π.

After expressing the Hamiltonian in qubit operators, a set of gateoperations used to perform the calculation may be provided. Such amethod may comprise an order of gate operations implementable by aquantum computer. Implementation of such a set of operations may betailored specifically to the type of quantum computer and to the type ofcalculation to be performed. Disclosed herein are various examples forsolving an electronic structure problem.

Ising and/or QUBO-Type Reduced Hamiltonian:

A method of solving an electronic structure problem using a quantumannealer is provided. In such an example, the mean-field Hamiltonian maybe transformed into an Ising model. In some embodiments, the Ising modelmay be further transformed into a quadratic unconstrained boundaryoptimization (QUBO) problem. In some embodiments, any binaryoptimization problem may be solved. Such problems may be performed on aquantum annealer such as D-Wave; however, methods of the presentdisclosure may also be performed on universal gate quantum computers. Aquantum annealer may be limited to the operation of X (or Y) and Zgates. Future quantum annealers may be able to simulate the actions ofother gates, so the use of X and Z gates only is not considered to belimiting. Quantum annealers may be able to solve the ground state energyof a qubit Hamiltonian that includes Pauli X, Y, and Z terms which suchHamiltonian is sometimes referred to as a non-stochastic Hamiltonian.

The qubit mean-field Hamiltonian described above may assume the generalform of a high order Ising model. In such a transformation, the X, Y,and Z terms may be transformed to mean-field Ising terms. Methods suchas the parameterization of the spin-½ Hilbert space in spin coherentstates can be used to perform transformation. In some embodiments, afunctional is proposed herein based on using the spherical coordinatesof the Bloch sphere associated with the spin J of a single particle. Insome cases, a quantum computer may comprise qubits which each havesingle particle spin, thus in a qubit mean-field Hamiltonian, X, Y, andZ terms can be represented as functions of Z operators and theirspherical rotations. In some cases, the single particle spin is J=½. Insome embodiments, X→cos φ sin θ, Y→sin φ sin θ, and Z→cos θ Z, where thedomains of θ and φ are [0,π/2] and [0,2π] respectively. In otherembodiments, additional transformations can be performed, such that:X_(i)→cos φ_(i) sin θ_(i) Z_(j), Y_(i)→sin φ_(i) sin θ_(i) Z_(k), andZ_(i)→cos θ_(i) Z_(i), where additional Z terms are introduced throughthe transformation of the X and Y terms into Ising terms. In this case,the domains of θ and φ are now [0,π/2] and [0,π/2] respectively. In someembodiments, it may be beneficial to transform X into the quantum meanfield and add the additional Ising terms, while transforming Y into thequantum mean field, but not adding the additional Ising terms. In othercases the reverse may be true. Transforming the Pauli Z term into thequantum mean field with the Ising term Z may be used over othertransformations in many quantum mean field cases.

In some cases, parameterization of the spin-½ Hilbert space in spincoherent states may be followed by domain folding techniques, asdescribed elsewhere herein, for example in the section entitled “DomainFolding Approach”. For example, the following transformation isperformed with relation to the cluster amplitudes τ:

sin(2τ_(i))→Z _(j) sin(2τ_(i))

(1−cos(2τ_(i)))→(1−Z _(k) cos(2τ_(i)))

Where the Z terms are new spin variables (Pauli Z operators) and theirindices j, k are j=number of qubits in QMF Hamiltonian+i and k=j+i. Withthis transformation the domain for the cluster amplitude τ is reduced to[0,π/2]. In some embodiments, the first transformation may be performedon the Ising Hamiltonian, in which case the domain of the clusteramplitude is [0,π].

In some cases, higher than second order Pauli Z terms in the Hamiltonianmay be identified. The qubit Hamiltonian produced by the Jordan-Wigneror Bravyi-Kitaev transformation may contain two or more Pauli Z, X, Y,or I terms. When the transformation in the previous paragraph isapplied, the number of Pauli Z terms in a summation may be greater than2. In this case the Hamiltonian is identified as higher than secondorder. In this case the Pauli Z operator can be reduced to be a variablethat has the two possible values of {-1,1} since these are the twoeigenvalues of the Pauli Z matrix.

Additionally, appropriate substitutions may be developed using auxiliaryterms to produce a quadratic Ising model. In this case, Ising summationterms which have more than 2 Ising terms (more than 2 Pauli Z termsmultiplied together), an auxiliary term can be used. In one example, thehigher order term contains z₁z₂z₃, so the substitution of z₄=z₂z₃ isperformed which yields a new term, z₁z₄, where the condition z₄=z₂z₃ isheld for all used results of the annealing process. In this example, apenalty (e.g. cost) function may be added into the Hamiltonian whichwould hold the z₄=z₂z₃ term. These steps may also be referred to asreduction and penalization steps. The Ising or QUBO model may be reducedand penalized by the same method as above.

In an example, if the qubit Hamiltonian is H=hX₀Y₁Z₂, where 0, 1, and 2are arbitrary qubit indices, the subsequent reduced Hamiltonian wouldbe: H=h*cos(φ₀)sin(θ₀)sin(φ₁)sin(θ₁)cos(θ₂)Z₂. The method removes the Xand Y Pauli operators completely from the expression, so Z₂ operatorsremain. The may take advantage of the fact that the QPU may not be ableto perform true X or Y measurements. In this specific example, if the Xand Y terms were to be transformed such that X_(i)→cos φ_(i) sin θ_(i)Z_(j), Y_(i)→sin φ_(i) sin θ_(i) Z_(k), the resulting Hamiltonian wouldbe: H=h*cos(φ₀)Z₃ sin(θ₀)sin(φ₁)Z₄ sin(θ₁)cos(θ₂)Z₂. This may takeadvantage of other symmetries in the quantum mean field Hamiltonian sothat the time to solve the problem is reduced by reducing the size ofthe domain of φ.

A specific example for 4 qubit H₂ is as follows. Pre-transformation theHamiltonian may be proportional to the following:

HαZ ₀ +Z ₀ +Z ₁ +Y ₀ Z ₁ Y ₂ +X ₀ Z ₁ X ₂ +X ₀ Z ₁ X ₂ Z ₃ +Y ₀ Z ₁ Y ₂Z ₃ +Z ₂ +Z ₀ Z ₂ +Z ₁ Z ₂ Z ₃ +Z ₀ Z ₁ Z ₂ Z ₃ +Z ₀ Z ₁ Z ₂ +Z ₀ Z ₂ Z₃ +Z ₁ Z ₃

Post-transformation the reduced Hamiltonian may be proportional to thefollowing:

HαZ ₀ cos(θ₀)+Z ₀ cos(θ₀)Z ₁ cos(θ₁)+Z ₁ cos(θ₁)+sin(φ₀)sin(θ₀)Z ₁cos(θ₁)sin(φ₂)sin(θ₂)+cos(φ₀)sin(θ₀)Z ₁cos(θ₁)cos(φ₂)sin(θ₂)+cos(φ₀)sin(θ₀)Z ₁ cos(θ₁)cos(φ₂)sin(θ₂)Z ₃cos(θ₃)+sin(φ₀)sin(θ₀)Z ₁ cos(θ₁)sin(φ₂)sin(θ₂)Z ₃ cos(θ₃)+Z ₂ cos(θ₂)+Z₀ cos(θ₀)Z ₂ cos(θ₂)+Z ₁ cos(θ₁)Z ₂ cos(θ₂)Z ₃ cos(θ₃)+Z ₀ cos(θ₀)Z ₁cos(θ₁)Z ₂ cos(θ₂)Z ₃ cos(θ₃)+Z ₀ cos(θ₀)Z ₁ cos(θ₁)Z ₂ cos(θ₂)+Z ₀cos(θ₀)Z ₂ cos(θ₂)Z ₃ cos(θ₃)+Z ₁ cos(θ₁)Z ₃ cos(θ₃).

In this specific example, if the X and Y terms were to be transformedsuch that X_(i)→cos φ_(i) sin θ_(i) Z_(j), Y_(i)→sin φ_(i) sin θ_(i)Z_(k), the Post-transformation Hamiltonian may be proportional to thefollowing:

HαZ ₀ cos(θ₀)+Z ₀ cos(θ₀)Z ₁ cos(θ₁)+Z ₁ cos(θ₁)+sin(φ₀)Z ₄ sin(θ₀)Z ₁cos(θ₁)sin(φ₂)Z ₅ sin(θ₂)cos(φ₀)Z ₆ sin(θ₀)Z ₁ cos(θ₁)cos(φ₂)Z ₇sin(θ₂)+cos(φ₀)Z ₆ sin(θ₀)Z ₁ cos(θ_(i))cos(φ₂)Z ₇ sin(θ₂)Z ₃cos(θ₃)+sin(φ₀)Z ₄ sin(θ₀)Z ₁ cos(θ₁)sin(φ₂)Z ₅ sin(θ₂)Z ₃ cos(θ₃)+Z ₂cos(θ₂)+Z ₀ cos(θ₀)Z ₂ cos(θ₂)+Z ₁ cos(θ₁)Z ₂ cos(θ₂)Z ₃ cos(θ₃)+Z ₀cos(θ₀)Z ₁ cos(θ₁)Z ₂ cos(θ₂)Z ₃ cos(θ₃)+Z ₀ cos(θ₀)Z ₁ cos(θ₁)Z ₂cos(θ₂)+Z ₀ cos(θ₀)Z ₂ cos(θ₂)Z ₃ cos(θ₃)+Z ₁ cos(θ₁)Z ₃ cos(θ₃).

Implementation of the Reduced Hamiltonian on the Quantum Computer

On a quantum computer, an initial estimate of the angle of each qubitmay be provided. Next, the interaction terms may be provided based on acalculation of the one and two electron integrals in the trialHamiltonian and the quantum computer initialized. This calculation maycomprise the bias and coupling terms for the qubits in the reducedHamiltonian. This procedure may comprise, for example, use of avariational quantum eigensolver or phase estimation; however, anyprocedure for preparing trial states may be used. In some cases,embedding the Hamiltonian on the quantum computer may comprise settingthe bias and coupling terms of individual qubits (for example, byapplying an appropriate magnetic field). In the case of a quantumannealer, the Hamiltonian may be physically embedded.

After each trial state is prepared, the associated energy may beestimated by measuring and summing the energy of the individual Pauliterms in the Hamiltonian. The energy estimates may then be used in agradient descent algorithm to optimize the control parameters. Thecontrol parameter may be minimization of the energy of a quantum state.Sufficient minimization may comprise a condition when changes in theenergy or the control parameter are below a threshold value.Additionally or alternatively, sufficient minimization may comprisehaving reached a predetermined number of iterations. Additionally oralternatively, sufficient minimization may comprise having reached apredetermined calculation time, such as a time at which the qubits havelost coherence.

FIG. 3 shows an example method 300 for implementing an Ising model on aquantum annealer. At an operation 310 of the method 300, the mean fieldHamiltonian in qubit space may be provided. At an operation 320 of themethod 300, higher than second order Pauli Z terms in the Hamiltonianmay be identified. Additionally, appropriate substitutions may bedeveloped using auxiliary terms to produce a quadratic Ising model. Atan operation 330 of the method 300, penalty functions may be enforcedinto h (bias) and j (coupling) terms for each qubit. The penaltyfunctions as well as the weight and the bias values for each qubit maybe evaluated on a classical computer of the present disclosure. At anoperation 340 of the method 300, the Ising model may be embedded on thequantum annealer. At an operation 350 the qubits may be sampled and anoutput recorded. Additionally, solutions may be selected which areconsistent with substitution and auxiliary qubits. Operations 330, 340,and 350 may be repeated at an operation 360 of the method 300 until aminimum a criterion is reached. The criterion may comprise a thresholdvalue. The criterion may comprise a minimum. The minimum may be a localminimum or a global minimum. The minimum may comprise a solution to theproblem.

FIG. 4 shows an example method 400 for implementing a QUBO model on aquantum annealer. At an operation 410 of the method 400, the mean fieldHamiltonian in qubit space may be provided. At an operation 420 of themethod 400, higher than second order Pauli Z terms in the Hamiltonianmay be identified. Additionally, appropriate substitutions may bedeveloped using auxiliary terms to produce a quadratic Ising model. Atan operation 430 of the method 400 the Ising model may be transformedinto a QUBO model. At an operation 440 of the method 400, penaltyfunctions may be enforced into h (bias) and j (coupling) terms for eachqubit. The penalty functions as well as the weight and the bias valuesfor each qubit may be evaluated on a classical computer of the presentdisclosure. At an operation 450 of the method 400, the QUBO model may beembedded on the quantum annealer. At an operation 460 the qubits may besampled and an output recorded. Additionally, solutions may be selectedwhich are consistent with substitution and auxiliary qubits. Operations430, 440, 450, and 460 may be repeated at an operation 470 of the method400 until a criterion is reached. The criterion may comprise a thresholdvalue. The criterion may comprise a minimum. The minimum may be a localminimum or a global minimum. The minimum may comprise a solution to theproblem.

Variational Approaches and Quantum Mean-Field Ansatz

In an aspect, a method of solving an electronic structure problem usinga qubit mean-field ansatz is provided. Such a method of solving aproblem may not require transforming the Hamiltonian into an Ising orQUBO type problem. Methods of the present disclosure may improve uponthe scaling of implementations of a variational quantum eigensolver forthe solution of electronic structure problems. Methods of the presentdisclosure may extend the quantum mean-field Hamiltonian to coupledcluster singles and doubles (CCSD). Methods comprising the qubit meanfield ansatz may be implemented on a universal gate quantum computer;however, methods of the present disclosure may be implemented on quantumand/or classical computers which simulate the action of the particulargate operations.

At an operation 255 of the method 200, a qubit mean-field ansatz may beprovided. In order to practically implement a method for solving anelectronic structure problem it may be beneficial to implement anoperation by step procedure which may be performed using a quantumcircuit (e.g. an ansatz). This quantum circuit may be expressed as a setof sequential gate operations. Such a procedure may be sufficientlyhardware efficient to perform a method of solving an electronicstructure problem within practical time frames, such as within a timeperiod that elapses prior to decoherence of entanglements.

Variational Approaches

In an example, one procedure may comprise the quantum phase estimationalgorithm. However, in some implementations, quantum computers whichimplement the quantum phase estimation algorithm may be susceptible toloss of coherence. In another example, a variational quantum eigensolverapproach may ameliorate such susceptibility and thereby facilitatecomputation. Example ansatzes provided herein may be implemented using avariational quantum eigensolver or the quantum estimation algorithm.

In a first example ansatz, n-qubit trail states may be parametrized as:

$ {{\Psi(\theta)} = {\prod\limits_{q = 1}^{n}\;{\lbrack {U^{q,d}(\theta)} \rbrack \times U^{ENT} \times \ldots\mspace{14mu} \times {\prod\limits_{q = 1}^{n}\;{\lbrack {U^{q,1}(\theta)} \rbrack \times U^{ENT} \times {\prod\limits_{q = 1}^{n}\;{\lbrack {U^{q,0}(\theta)} \rbrack\text{|}00\mspace{14mu}\ldots\mspace{14mu} 0}}}}}}} \rangle$

The parametrization may be similar to that disclosed in Kandala (A.Kadala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. chow, and J.M. Gambetta, Nature 549, 242 (2017)), which is herein incorporated byreference in its entirety. The above equation has a structure of analternating sequence of products of individual qubit rotationsU^(q,i)(θ),

U ^(q,i)(θ)=e ^(iz) ^(q) ^(θ) ¹ ^(q,i) e ^(ix) ^(q) ^(θ) ² ^(q,i) e^(iz) ^(q) ^(θ) ³ ^(q,i), 0≤i≤d

where θ₁ ^(q), θ₂ ^(q), and θ₃ ^(q) are the Euler angles of the q-thspin, interleaved with action of “entanglers” U_(ent)=exp(−iτĤ_(o)),where τ is the fixed amplitude and Ĥ₀ is a multi-qubit operator (i.e. anentanglement gate comprising at least two qubits). The number dcomprises a depth scheme or ansatz depth. The depth scheme defines anumber of iterations over which the entanglement scheme may be repeated.The depth scheme may comprise a number of iterations sufficient to reachchemical accuracy.

In the above example ansatz, a computational work flow may comprise Xand Z rotation gates (by the rotations in U^(q,i)) and an entanglementgate. The Ansatz of Kandala et al. comprises n*(3d+2) gate operations intotal. Such a work flow may reach chemical accuracy with an ansatz depthof 16.

Quantum Mean-Field Ansatz:

In an additional example ansatz, it may be beneficial to reduce thenumber of qubit operations used to reach chemical accuracy for similarquantum computers. Such an ansatz may be implemented by adopting analternative parameterization of the Hilbert space of an individualqubit. Such an alternative parameterization of the Hilbert space maycomprise spin coherent states. Spin coherent states may parameterize theHilbert space in so-called “Bloch states” defined by a raising orlowering spin operator acting on a particle with spin J in sphericalpolar coordinates, rather than Euler angles. The energy of ananalogously presented Hamiltonian, using such a parametrization, wasfound to be an upper bound to the exact energy. See, for instance, Lieb(E. H. Lieb, Commun. Math. Phys. 31, 327 (1973)), which is hereinincorporated by reference in its entirety.

Direct product of states |Ω> defined elsewhere herein provides a basisfor an n-qubit system and the ground state energy is bound from aboveby:

${E_{0} \leq \langle {\Omega{\hat{H}}\Omega} \rangle} = {\sum\limits_{I}{C_{I}{F_{I}( {n_{1}^{\omega},{\ldots\mspace{14mu} n_{n}^{\omega}}} )}}}$$E_{QMF} = {\sum\limits_{I}{C_{I}{F_{I}( {n_{1}^{\omega},{\ldots\mspace{14mu} n_{n}^{\omega}}} )}}}$

where the right hand side defines the qubit mean-field energy functionaland where each F_(I) is obtained from T_(I) by substitution ofω_(i)→jn_(i) ^(ω). Operator products of ω_(i) are converted to ordinarynumerical products. n_(i) ^(ω) is shorthand for ω component of the unitvector on a Bloch sphere: n=(cos ϕ sin θ, sin ϕ sin θ, cos θ)→(x_(i),y_(i), z_(i)).

As shown, the qubit mean-field Ansatz comprises X and Z rotation gatesin n without additional entanglement gates or an Ansatz depth parameter(d) of Kandala et al. As such, it also makes sense to call the qubitmean-field Ansatz an independent-qubit model. The quantum mean-fieldAnsatz may implemented similarly to the variational quantum eigensolverof Kandala et al. where effectively d=0. Implementation of the quantummean field Ansatz scales with 2*n gates (i.e. linearly).

Coat or Penalty Functionals:

Quantum computers, quantum annealers, and simulated annealing techniquesmay have noise associated with them. Among other causes, noise may causemeasurement of the qubits to be inaccurate. In some cases, one or morequbits may be incorrectly measured as a result of noise such as readinga qubit that should be 0 as 1 instead. The variational quantumeigensolver (VQE) algorithm may be susceptible to this noise and mayresult in the expectation value being higher than the energy ofinterest. In other cases, it may be difficult to get the VQE algorithmto get the energy of a selected molecular state (such as the tripletstate or the first excited state). As an optional step of any aspect ofthe methods described herein, it may be beneficial to transform theQubit Hamilton of the system of interest by adding a penalty functional.In some cases, this transformation may occur prior to generating thequbit mean field or qubit coupled cluster ansatz and/or performing atransformation into an Ising Hamiltonian.

In some embodiments, it may be beneficial to constrain minimization ofthe qubit Hamiltonian with respect to a penalty functional. Spinoperators, such as, for example, the square of the total spin (S²) andthe projection of the total molecular spin (S_(z)) may be used.Additionally or alternatively, number operators ({circumflex over(N)}=Σ_(i) a_(i) ^(†)a_(i)) may be used. Additionally or alternatively,other operators which commute with the Hamiltonian may be used as aconstraint. In some cases, the commutator itself may be used as aconstraint.

An example implementation of a constraint on the total spin operator isas follows:

${ɛ_{s}( {\Omega_{1},\ldots\mspace{14mu},\Omega_{4},\mu} )} = {{E_{QMF}( {\Omega_{1},\ldots\mspace{14mu},\Omega_{4}} )} + {\frac{\mu}{2}\lbrack {{S_{QMF}^{2}( {\Omega_{1},\ldots\mspace{14mu},\Omega_{4}} )} - S^{2}} \rbrack}^{2}}$

where the total spin is constrained to be zero and where μ is a penaltyparameter which greater than 1.

Alternatively, a constraint on the number operator may be implemented asfollows:

${ɛ_{N}( {\Omega_{1},\ldots\mspace{14mu},\Omega_{4},\mu} )} = {{E_{QMF}( {\Omega_{1},\ldots\mspace{14mu},\Omega_{4}} )} + {\frac{\mu}{2}\lbrack {{N_{QMF}( {\Omega_{1},\ldots\mspace{14mu},\Omega_{4}} )} - 1} \rbrack}^{2}}$

Because these operators share the same Pauli terms as the Hamiltonian,the values of these operators may be calculated without additional orminimal computational cost.

FIG. 19 shows an example method 1900 of implementing a constraint usingprojectors. The method 1900 may be used to remove selected spin ornumber states from the Qubit Hamiltonian. At an operation 1910, aHamiltonian in qubit/spin space may be provided. At an operation 1920,the number and spin operators may be generated. At an operation 1930, anexpression for the qubit Hamiltonian based on the operator and thetargeted eigenvalue of the operator may be derived. The number and spinoperators in qubit space may be composed of Pauli Word Operatorscomprising of products of Pauli X, Y, and Z spin operators. Theseoperators may be generated by using the Bravyi-Kitaev or Jordan-Wignertransformation on their fermionic spin and number operators as describedelsewhere herein. In these cases, the number operator and spin operatorsin Qubit/Spin space both commute with the Qubit Hamiltonian. Since theseoperators commute with the Qubit Hamiltonian, they possess a common setof eigenfunctions, |ψ_(i)). The eigendecomposition of the QubitHamiltonian H can be described as:

$H = {\sum\limits_{i}{E_{i} \psi_{i} \rangle\langle \psi_{i} }}$

where E_(i) are the energy eigenvalues. At an operation 1940, aconstrained Qubit Hamiltonian may be derived. The Qubit Hamiltonian maybe constrained by a given operator and a given eigenvalue. This resultsin the general case, where the Qubit Hamiltonian H_(ai) is:

H _(ai) =H−H(A−a _(i))²−(A−a _(i))² H

Where A is the operator of interest (for example, S₂, N, etc.) in Qubitform, and a_(i) is the target eigenvalue of the operator A. In thiscase, the constrained Qubit Hamiltonian is generally larger. The form ofthe operator (A−a_(i))² is quadratic, but in some cases it does not haveto be quadratic. At an operation 1950, operations 1930 and 1940 may berepeated. At an operation 1960, the constrained Hamiltonian may beembedded on the quantum computer. The quantum computer may be a quantumannealer or a universal gate quantum computer.

In an example using the spin operator, it can be applied onto the aboveequation such that:

S ² H=Σ _(i) S _(i)(S _(i)+1)E _(i)|ψ_(i)

ψ_(i) |, S=0,1, . . .

If the singlet state is selected in such that S=0, a new QubitHamiltonian can be described as:

H _(S=0) =H−HS ² −S ² H

And thus the eigendecomposition of the above equation would be thefollowing:

$H_{S = 0} = {{\sum\limits_{i}{E_{i} \psi_{i} \rangle\langle \psi_{i} }} + {\sum\limits_{i}{{- {E_{i}\lbrack {{2{S_{i}( {S_{i} + 1} )}} - 1} \rbrack}} \psi_{i} \rangle\langle \psi_{i} }}}$

The above equation may cause eigenstates of the Qubit Hamiltonian thatare not singlets (the state of interest), to be shifted to positiveenergies for the ground state. This may enable VQE style algorithms tohave a higher accuracy, since they sample from nearby negative energystates. The constrained Qubit Hamiltonian may be run on a universal gatequantum computer, by composing the quantum mean-field (QMF) or qubitcoupled-cluster (QCC) ansatz. Additionally or alternatively, theconstrained Qubit Hamiltonian can be transformed into the Ising form. Insome embodiments, the Qubit Hamiltonian can be acted upon by multipleoperators, one at a time, in such that spin and number are constrained.In some embodiments, to compensate for the increase in Pauli Word Sumsafter the transformation, the new Qubit Hamiltonian may be processedthrough a greedy algorithm to determine which Pauli Words have commontensor product basis decomposition.

Qubit Coupled Cluster

In some embodiments, the quantum mean field Ansatz may also be extendedto coupled cluster singles and doubles, or to the full configurationinteraction with the addition of entanglement gates. In an example, atan operation 265 of the method 200, the qubit mean-field Ansatz may beextended to CCSD by adding entanglement scheme using XZ gates. The qubitmean-field Hamiltonian may be further transformed by acting from theleft and right with the entanglement operator U_(ent)=exp(−iτĤ_(o)),where τ is the fixed amplitude and Ĥ₀ is a multi-qubit operator (i.e. anentanglement gate comprising at least two qubits).

Ĥ(τ)=e ^(iτ{circumflex over (T)}) Ĥe ^(−iτ{circumflex over (T)})

This transformation comprises the qubit coupled-cluster Ansatz. Usingsuch a transformation, the reference state Ω and the “cluster amplitude”τ may be variationally optimized together.

A reduced energy functional, based on the general Unitary Ansatz appliedon qubit mean-field Hamiltonian may be defined as below:

${E_{cQMF}^{\hat{T}}(\tau)} = {\min\limits_{\Omega}\langle {\Omega{{e^{i\;\tau\hat{T}}\hat{H}e^{{- i}\;\tau\hat{T}}}}\Omega} \rangle}$

The above is a function of an amplitude τ and a generator {circumflexover (T)}, where 0≤τ<π/2. For example, a generator may compriseinvoluntary products of two or more Pauli matrices. The generator may beof the form x_(P)x_(R), x_(P)z_(R), z_(P)z_(R), x_(P)y_(R), y_(P)z_(R),y_(P)y_(R), z_(P)z_(R), etc. where P and R are qubit indices and P doesnot equal R. The amplitude τ may comprise a rotation of the entanglementoperator. The rotation of the entanglement operator may arise from asequence of gate operations which may be used in the entanglementscheme, for example: Hadamard, CNOT, RZ, CNOT, Hadamard.

FIG. 13A and FIG. 13B show an example entanglement ansatz scheme 1300based on alpha-beta molecular orbital blending. The entanglement ansatzscheme may be based on the composition of alpha-beta molecular orbitalblending schemes. In such an entanglement ansatz scheme, the perceivedalpha and beta molecular orbitals of the molecule are mapped onto thequbits, where each qubit or a combination of qubits represents an alphaor beta spin molecular orbital. For example, alternating qubits may berepresented as alpha and beta spin orbitals, such that a beta qubit isarranged in between each neighboring alpha qubits and vice versa. Eachalpha qubit may be entangled with at least one other alpha qubit, andeach beta qubit may be entangled with at least one other beta qubit. Forexample, as shown in example 1310, each alpha qubit may be entangledwith at least one adjacent alpha qubit and each beta qubit may beentangled with at least one adjacent beta qubit. The nearby orbitals aremolecular orbitals which are directly above or below the current orbitalin question based on energy value. In some cases, entanglement ansatzschemes are based on entangling appropriate sigma and pi bondingorbitals. For example, as shown in example 1300, each alpha qubit may beentangled with at least one alpha qubit and each beta qubit may beentangled with at least one beta qubit, where the qubits are notadjacent. In another entanglement scheme, as shown in 1320, theentangler can be selected based on the rankings of Pauli word operators,which in this case, results in x₂y₀ entangler. The Pauli word operatorsmay be decomposed into a set of logical circuit quantum gates thatinvolves CNOT, RZ, RX and Hademard gates. The exact selection of thequantum circuit gates may be modified based on hardware and what thenative set of gates is. For a Universal quantum computer that has nativeRX, RZ, CNOT, Hademard, gates, the Pauli word entangler x₂y₀ can bedecomposed into a set of these gates as shown in 1330.

Ranked Qubit Coupled Cluster

In an additional or alternative scheme of extending the qubit mean-fieldAnsatz, an additional step of the method may comprise determiningentanglement scheme based on ranking possible entanglers based on theircontribution to the overall correlation energy. In such a method, thegenerator {circumflex over (T)} can be composed of Pauli words. In somecases, Pauli words may be selected so as to avoid breaking fermionicsymmetry. To determine which entanglers and generators to use, thegenerators which minimize the individual Qubit coupled cluster (QCC)energy function

${E\lbrack {\tau;\hat{P_{k}}} \rbrack} = {\min\limits_{\Omega}\langle {\Omega{{e^{i\;\tau\hat{P_{k}}}\hat{H}e^{{- i}\;\tau\hat{P_{k}}}}}\Omega} \rangle}$

may be used, where P_(k) represents the Pauli generator and |Ω

=Π_(i) ^(Nent)|Ω_(i)

represents the mean-field portion of the wavefunction indexed by eachi^(th) qubit.

In some examples, the above equation can be expanded via a Taylor seriesexpansion to a new form, which generates the three term equation whentruncated at second order:

${{{{{E\lbrack {\tau;\hat{P_{k}}} \rbrack} = {E_{QMF} + {\tau\frac{d{E\lbrack {\tau;\hat{P_{k}}} \rbrack}}{d\tau}}}}}_{\tau = 0} + {\frac{\tau^{2}}{2}\frac{d^{2}{E\lbrack {\tau;\hat{P_{k}}} \rbrack}}{d\tau^{2}}}}}_{\tau = 0}$

With the above equation, the second term and third term can be evaluatedto determine whether a given combination of Pauli words is acting as anentangler.

In an example, entanglers which result in greater than 0.001 a.u.absolute value in the first derivative may be considered to be acting asan entangler. In another example, a Pauli word may be acting as anentangler if the absolute value of the first derivative is greater than0.01 a.u., greater than 0.001 a.u., greater than 0.0001 a.u., greaterthan 0.0001 a.u., or greater than 0.00001 a.u. In another example,entanglers which have an absolute value <0.001 a.u., but have a negativesecond energy derivative may also be considered significant. In somecases, the method may comprise searching for any length of Pauli wordwhich may act as an entangler. In some embodiments, the length of thePauli word that may be screened using this process may be limited inlength to 2, 3, 4, etc. Pauli spin operators.

The entanglers indexed by their contribution to the overall correlationenergy may be integrated into the QCC Hamiltonian as follows. For anumber of entanglers k, the correlation part of the QCC wavefunctionU(τ) can be represented as Û(τ)=Π_(k) ^(Nent)exp(−iτ_(k){circumflex over(P)}_(k)) where P_(k) are the Pauli words whose length can vary from twoto the number of qubits and T_(k) represents a real-valued amplitude ofthe multi-qubit rotation. The number of entangler in the QCC Hamiltonianmay vary from 1 to 4^(N) ^(q) −2N_(q)−1. The expectation value of theQCC Hamiltonian may then be E(τ, Ω)=

Ω|Û(τ)^(†)ĤÛ(τ)|Ω> and minimization of the expectation value withrespect to the rotation amplitudes and angles in Ω yields the groundstate energy.

In examples where the Pauli word is a product of more than two qubits,the unitary transformation comprising the Pauli word may be factoredinto products of unitary transformations comprising two qubits. Theprocedure may recursively produce three new entanglers each containingfewer qubits than the original Pauli word. The factorization proceduremay be summarized as follows. Assume Pauli word with length greater thanor equal to 3 may be assumed, where the length is the number of Paulioperators in P. The Pauli word to be factored can be represented as{circumflex over (P)}={circumflex over (P)}₁ω_(i){circumflex over (P)}₂where ω_(i) represents the elementary Pauli operator corresponding tothe i-th qubit. The word P1 (P2) may contain all qubit indices that arestrictly greater (lower) than i. Subsequently, a commutation relationmay be defined as follows: [{circumflex over (ω)}′_(i), {circumflex over(ω)}″_(i)]=2i{circumflex over (ω)}_(i). Substituting the relation intothe Pauli word yields the relation

${\hat{P} = {- {{\frac{1}{2}\lbrack {{{\hat{P}}_{1}{\hat{\omega}}_{i}^{\prime}},{{\hat{\omega}}_{i}^{''}{\hat{P}}_{2}}} \rbrack}.\mspace{14mu}{Therefore}}}},{\hat{P} = {e^{{i{(\frac{\pi}{4})}}{\hat{\omega}}_{i}^{''}{\hat{P}}_{2}}{\hat{P}}_{1}{\hat{\omega}}_{i}^{\prime}e^{{- {i{(\frac{\pi}{4})}}}{\hat{\omega}}_{i}^{''}{\hat{P}}_{2}}}},{{{and}\mspace{14mu}{\hat{P}}_{n}} = {{e^{{i{(\frac{\pi}{4})}}{\hat{\omega}}_{i}^{''}{\hat{P}}_{2}}( {{\hat{P}}_{1}{\hat{\omega}}_{i}^{\prime}} )}^{n}{e^{{- {i{(\frac{\pi}{4})}}}{\hat{\omega}}_{i}^{''}{\hat{P}}_{2}}.}}}$

Subsequently, exponentiation of P and Taylor expansion yields thefactorization

$e^{{- {it}}\hat{P}} = {e^{{i{(\frac{\pi}{4})}}{\hat{\omega}}_{i}^{''}{\hat{P}}_{2}}e^{{- {it}}{\hat{P}}_{1}{\hat{\omega}}_{i}^{\prime}}{e^{{- {i{(\frac{\pi}{4})}}}{\hat{\omega}}_{i}^{''}{\hat{P}}_{2}}.}}$

FIG. 14 shows an example method 1400 comprising ranking the Paulioperators by their contribution to the QCC Hamiltonian. At an operation1410 of the method 1400, a molecular Hamiltonian in Qubit/Spin space maybe generated. At an operation 1420 of the method 1400, the QCCHamiltonian may be Taylor expanded to include first and secondderivative terms. At an operation 1430 of the method 1400, potentialPauli Word entanglement Operators may be generated based on the numberof qubits. At an operation 1440 of the method 1400, the first and secondderivatives may be evaluated. At an operation 1450 of the method 1400,Pauli Word entanglement operators may be selected. These operators maybe those which have non-zero energy values in the first derivative. Insome cases, the number of entanglement operators may be truncated. At anoperation 1460 of the method 1400, for those entanglement operatorswhich have close to zero first derivative (e.g. less than 0.001 a.u.absolute value as described herein), those which have a negative secondderivative may be selected. At an operation 1470 of the method 1400, theentanglement operators may be ranked based on which ones contribute mostto the change in the total energy. In some cases, entanglement operatorsmay be factored into two-qubit operators. At an operation 1480 of themethod 1400, an entanglement ansatz may be generated based on theselected entanglement operators.

FIG. 16 shows a method 1600 comprising ranking the Pauli operators bytheir contribution to the QCC Hamiltonian and implementing the QCCmethod on a quantum annealer. In some cases, the entanglement Ansatz forH₂ acting upon 4 qubits may be determined using the Pauli word generatormethod as described above. In some cases, the Pauli generator may bechosen to be x₂y₀. This results in the following QCC Hamiltonian:

Ĥ(τ;x ₂ y ₀)=e ^(iτx) ² ^(y) ⁰ Ĥe ^(−iτx) ² ^(y) ⁰

The above Hamiltonian has a similar form to Unitary Coupled ClusterAnsatz. Here the above equation may be transformed into an equationusing the method described herein into a new Hamiltonian:

${{\hat{\mathcal{H}}( {\tau;{x_{2}y_{0}}} )} = {{\hat{H}}_{QMF} + {\frac{\sin 2\tau}{2}( {- {i\lbrack {\hat{H},{x_{2}y_{0}}} \rbrack}} )} + {\frac{1}{2}( {1 - {\cos 2\tau}} )x_{2}{y_{0}\lbrack {\hat{H},{x_{2}y_{0}}} \rbrack}}}},$

where the variable Ĥ_(QMF) is the quantum is mean field energy of theHamiltonian and τ is a cluster amplitude that that may be iterativelyreduced until a threshold condition may be reached. In this example, thesummation of 4 qubit Pauli words which describe −i[Ĥ, x₂y₀] isproportional to: −i[Ĥ, x₂y₀]αZ₀Z₁+X₀X₂+X₀Z₁X₂+Y₀ Y₂+Z₁Z₂ Z₀Z₁Z₃ Y₀Z₁Y₂Z₃Z₁Z₂Z₃ and x₂y₀[Ĥ,x₂y₀] is represented by the following expression:

x ₂ y ₀[Ĥ,x ₂ y ₀]αZ ₀ +Z ₀ Z ₁ +X ₀ Z ₁ X ₂ +Y ₀ Z ₁ Y ₂ +Z ₂ +X ₀ Z ₁X ₂ Z ₃ Y ₀ Z ₁ Y ₂ Z ₃ Z ₁ Z ₂ Z ₃.

These expressions may be both be transformed into an Ising model usingthe methods described herein above by transforming the Pauli Z, X and Yoperators. This may result in the following expressions:

−i[Ĥ, x₂y₀]αZ₀cos θ₀Z₁cos θ₁ + sin  φ₀cos θ₀sin  φ₂cos θ₂ + sin  φ₀cos θ₀Z₁cos θ₁sin  φ₂cos θ₂ + sin  φ₀sin θ₀sin  φ₂sin θ₂ + Z₁cos θ₁Z₂cos θ₂ + Z₀cos θ₀Z₁cos θ₁Z₃cos θ₃ + sin  φ₀sin θ₀Z₁cos θ₁sin  φ₂sin θ₂Z₃cos θ₃ + Z₁cos θ₁Z₂cos θ₂Z₃cos θ₃

And:

x₂y₀[Ĥ, x₂y₀]αZ₀cos θ₀ + Z₀cos θ₀Z₁cos θ₁ + sin  φ₀cos θ₀Z₁cos θ₁sin  φ₂cos θ₂ + sin  φ₀sin θ₀Z₁cos θ₁sin  φ₂sin θ₂ + Z₂cos θ₂ + sin  φ₀cos θ₀Z₁cos θ₁sin  φ₂cos θ₂Z₃cos θ₃ + sin  φ₀sin θ₀Z₁cos θ₁sin  φ₂sin θ₂Z₃cos θ₃ + Z₁cos θ₁Z₂cos θ₂Z₃cos θ₃

In some embodiments the Pauli X and Y operators can be transformedfurther, resulting in:

−i[Ĥ, x₂y₀]αZ₀cos θ₀Z₁cos θ₁ + sin  φ₀Z₄cos θ₀sin  φ₂Z₅cos θ₂ + sin  φ₀Z₄cos θ₀Z₁cos θ₁sin  φ₂Z₅cos θ₂ + sin  φ₀Z₄sin θ₀Z₆sin  φ₂Z₅sin θ₂Z₇ + Z₁cos θ₁Z₂cos θ₂ + Z₀cos θ₀Z₁cos θ₁Z₃cos θ₃ + sin  φ₀Z₄sin θ₀Z₆Z₁cos θ₁sin  φ₂Z₅sin θ₂Z₇Z₃cos θ₃ + Z₁cos θ₁Z₂cos θ₂Z₃cos θ₃

And:

x₂y₀[Ĥ, x₂y₀]αZ₀cos θ₀ + Z₀cos θ₀Z₁cos θ₁ + sin  φ₀Z₄cos θ₀Z₁cos θ₁sin  φ₂Z₅cos θ₂ + sin  φ₀Z₄sin θ₀Z₆Z₁cos θ₁sin  φ₂Z₅sin θ₂Z₇ + Z₂cos θ₂ + sin  φ₀Z₄cos θ₀Z₁cos θ₁sin  φ₂Z₅cos θ₂Z₃cos θ₃ + sin  φ₀Z₄sin θ₀Z₆Z₁cos θ₁sin  φ₂Z₅sin θ₂Z₇Z₃cos θ₃ + Z₁cos θ₁Z₂cos θ₂Z₃cos θ₃

With the above transformations, the ground state energy of a molecularHamiltonian in spin basis derived from the qubit coupled cluster theorymay be solved using the Ising formulation.

FIG. 16 shows an example method comprising ranking the Pauli operatorsby their contribution to the QCC Hamiltonian and implementing the QCCmethod on a quantum annealer. At an operation 1610 of the method 1600, amolecular Hamiltonian in Qubit/Spin space may be generated. At anoperation 1615 of the method 1600, the QCC Hamiltonian may be Taylorexpanded to include first and second derivative terms. At an operation1620 of the method 1600, a potential Pauli Word entanglement Operatorsmay be generated based on the number of qubits. At an operation 1625 ofthe method 1600, the first and second derivatives may be evaluated. Atan operation 1630 of the method 1600, Pauli Word entanglement operatorsmay be selected. These operators may be those which have non-zero energyvalues in the first derivative. In some cases, the number ofentanglement operators may be truncated. At an operation 1635 of themethod 1600, for those entanglement operators which have close to zerofirst derivative, those which have a negative second derivative may beselected. At an operation 1640 of the method 1600, the entanglementoperators may be ranked based on which ones contribute most to thechange in the total energy. In some cases, entanglement operators may befactored into two-qubit operators. At an operation 1645 of the method1600, an entanglement ansatz may be generated based on the selectedentanglement operators.

At an operation 1650 of the method 1600, the x, y, and z terms may beconverted to Ising terms using a Bloch rotation. At an operation 1655 ofthe method 1600, the Hamiltonian may be transformed into an Ising formusing a Bloch sphere rotation substitution. At an operation 1660 of themethod 1600, a qubit coupled cluster Hamiltonian in qubit/spin space maybe generated. At an operation 1665 of the method 1600, the higher thansecond order Pauli z terms in the Hamiltonian may be identified. In somecases, auxiliary terms may be substituted into the Hamiltonian toproduce a quadratic Ising model. At an operation 1670 of the method1600, the Ising model may be transformed into a QUBO model. At anoperation 1675 of the method 1600, penalty functions may be includedinto the bias (h) and coupling (j) terms. These terms may be calculatedon a CPU. At an operation 1680 of the method 1600, the Ising model maybe embedded on a quantum annealer. At an operation 1685 of the method1600, the output qubits may be sampled and read. Solutions may beselected which are consistent with substitution and auxiliary qubits. Atan operation 1690 of the method 1600, operations 1675 to 1685 may berepeated. Weight and bias terms may be varied using the CPU. In somecases, a local minimum may be reached. In some cases, a global minimummay be reached. In some cases the operations may be repeated until aselected threshold has been reached.

Domain Folding Approach

In addition to the above approaches including Ising style approaches, anadditional two folding procedures may be performed on the qubit coupledcluster Hamiltonian. A folding procedure may be beneficial in order toshorten convergence times. A folding procedure may be beneficial toreduce the number of local minima on the optimization surface. In somecases, the following transformation is performed with relation to thecluster amplitudes τ:

sin(2τ_(i))→Z _(j) sin(2τ_(i))

(1−cos(2τ_(i)))→(1−Z _(k) cos(2τ_(i)))

Where the Z terms are new spin variables (Pauli Z operators) and theirindices j, k are j=number of qubits in QMF Hamiltonian+i and k=j+i. Withthis transformation the domain for the cluster amplitude τ is reduced to[0,π/2]. In some embodiments, the first transformation may be performedon the Ising Hamiltonian, in which case the domain of the clusteramplitude is [0,π].

FIG. 18 shows an example method 1800 comprising ranking the Paulioperators by their contribution to the QCC Hamiltonian, folding theoptimization domain, and implementing the QCC method on a quantumannealer. At an operation 1810 of the method 1800, a molecularHamiltonian in Qubit/Spin space may be generated. At an operation 1815of the method 1800, the QCC Hamiltonian may be Taylor expanded toinclude first and second derivative terms. At an operation 1820 of themethod 1800, a potential Pauli Word entanglement Operators may begenerated based on the number of qubits. At an operation 1825 of themethod 1800, the first and second derivatives may be evaluated. At anoperation 1830 of the method 1800, Pauli Word entanglement operators maybe selected. These operators may be those which have non-zero energyvalues in the first derivative. In some cases, the number ofentanglement operators may be truncated. At an operation 1835 of themethod 1800, for those entanglement operators which have close to zerofirst derivative, those which have a negative second derivative may beselected. At an operation 1840 of the method 1800, the entanglementoperators may be ranked based on which ones contribute most to thechange in the total energy. In some cases, entanglement operators may befactored into two-qubit operators. At an operation 1845 of the method1800, an entanglement ansatz may be generated based on the selectedentanglement operators.

At an operation 1850 of the method 1800, the x, y, and z terms may beconverted to Ising terms using a Bloch rotation. At an operation 1855 ofthe method 1800, the Hamiltonian may be transformed into an Ising formusing a Bloch sphere rotation substitution. At an operation 1860 of themethod 1800, a qubit coupled cluster Hamiltonian in qubit/spin space maybe generated. At an operation 1865 of the method 1800, the higher thansecond order Pauli z terms in the Hamiltonian may be identified. In somecases, auxiliary terms may be substituted into the Hamiltonian toproduce a quadratic Ising model. At an operation 1870 of the method1800, the Ising model may be transformed into a QUBO model. At anoperation 1875 of the method 1800, the amplitudes and/or phases may befolded to reduce the domain space of amplitudes and/or phases. At anoperation 1880 of the method 1800, penalty functions may be includedinto the bias (h) and coupling (j) terms. These terms may be calculatedon a CPU. At an operation 1885 of the method 1800, the Ising model maybe embedded on a quantum annealer. At an operation 1890 of the method1800, the output qubits may be sampled and read. Solutions may beselected which are consistent with substitution and auxiliary qubits. Atan operation 1895 of the method 1800, operations 1875 to 1890 may berepeated. Weight and bias terms may be varied using the CPU. In somecases, a local minimum may be reached. In some cases, a global minimummay be reached. In some cases the operations may be repeated until aselected threshold has been reached.

Iterative Qubit Coupled Cluster

By way of summary, the Qubit Coupled Cluster (QCC) method may utilizeparametrization of the electronic wave function of a molecular systemas:

Ψ=Π_(k)exp(−it_(k) T _(k)/2)|Ω

, k=1, 2, . . .  (1)

where T_(k) are multi-qubit operators hereinafter termed “entanglers”,t_(k) are the corresponding numerical amplitudes (the first set ofcontrol variables), and |Ω> is a product of qubit coherent states|(φ_(j),θ_(j))>,

|Ω>=|(φ₁,θ₁)> . . . |(φ_(n),θ_(n))>,  (2)

where φ_(j),θ_(j) are Bloch angles for the i-th qubit; there are 2nBloch angles for a system with n qubits. Bloch angles constitute thesecond set of control variables.

In some cases, the form (1) can be directly implemented on a universalquantum computer as a quantum circuit. In some cases, the state |Ω> canbe prepared by acting of a set of single-qubit gates parametrized byvalues of Block angles; entanglers T_(k) are represented as multi-qubitentangling gates whose form may be selected by a user or by anyappropriate method, for example, in methods as described elsewhereherein. The entangling gates may be selected by at least twoconsiderations: i. which T_(k) provides the fastest convergence towardthe ground state energy (system-dependent) and ii. which T_(k) can beefficiently implemented as quantum gates with the lowest noise andlargest coherence times (hardware dependent). Once an acceptableselection of T_(k) is found, the corresponding quantum circuit may beused to execute multiple times on a quantum computer to produce Ψfollowed by measuring the expectation value of each single term (or somegroup of them) of the molecular Hamiltonian H_(e) to find out the meanvalue of the electronic energy

E _(e)(t _(k),φ_(j),θ_(j))=

Ψ|H _(e)|Ψ>.  (3)

The mean value of the electronic energy may be subsequently fed into anenergy optimizer running on a classical computer to minimize E_(e) withrespect to all control variables t_(k), φ_(j), θ_(j). According to ageneral Variational Quantum Eigensolver (VQE) scheme, which may providea variational upper bound for the ground-state energy of a molecularsystem with the Hamiltonian H_(e).

In some embodiments, the QCC procedure described herein above mayapplied once. In such cases, as long as the set of entanglers T_(k) ischosen and fixed and the electronic energy may be optimized with respectto all control variables, the value of the electronic energy may be theresult of electronic-structure calculations of the target system.Without being limited by theory, the challenge with this approach may befinding a functional trade-off between better (e.g. longer) sets ofentanglers T_(k) to achieve better (e.g. lower) ground-state energyversus the ability of a quantum computer to operate with the set ofchosen T_(k)-s.

In some embodiments, the QCC procedure described herein above may beapplied iteratively. In an iterative case, once the set of optimizedamplitudes t_(k)(1) is found (where the argument oft represents the stepat which they were obtained, starting from 1) the original electronicHamiltonian H_(e)(1) can be unitarily transformed in a step-wise manner:

H _(e)→exp(it₁(1)T ₁/2)H _(e) exp(−it₁(1)T ₁/2)→exp(it₂(1)T₂/2)exp(it₁(1)T ₁/2)H _(e) exp(−it₁(1)T ₁/2)exp(−it₂(01)T ₂/2)  (4)

At each step a closed expression exists for the transformed operator,for example:

(cos(t ₁/2)+i sin(t ₁/2)T ₁)H _(e)(cos(t ₁/2)−i sin(t ₁/2)=H _(e)+sin(t₁)/2(T ₁ H _(e) −H _(e) T ₁)+(1−cos(t ₁))/2(T ₁ H _(e) T ₁ −H _(e))  (5)

Such an operator product may be evaluated on a classical computer usingPauli polynomial manipulation software. After evaluation, the result isa new operator of the same structure as the original H_(e). In somecases, the result may contain more terms than H_(e). For example, if allthree terms in sum (5) are algebraically independent, one may expect3-fold increase in length (the number of terms) of the correspondingoperator. In contrast to expectation, the expansion may typically beless pronounced because summands [H_(e), (T₁H_(e)−H_(e)T₁),(T₁H_(e)T₁−H_(e))] share many common terms.

The steps of transforming the Hamiltonian and Euler expansion of thetransformed operator may be referred to as “dressing”. If all entanglersT_(k)(1) are exhausted in the dressing procedure (4) by successivelyapplying Eq. (5) along with the corresponding amplitudes, the resultingHamiltonian H_(e)(2) may be characterized by the mean value of E_(e)(1)that can be computed solely using the Bloch state |Ω(1)

without entangling gates:

E _(e)(1)≡

Ψ|H _(e) |Ψ

=

Ω|H _(e)(2)|Ω

  (6)

Moreover, since the sequence of transformations in Eq. (4) is unitary,H_(e)(2) may possess the same exact ground state as the original H_(e).Therefore, in some case, one may use H_(e)(2) in place of H_(e) as astarting point for the new iteration of the QCC procedure. Inparticular, one may find out a new set of entanglers T_(k)(2), convertthem into a quantum circuit, and run the optimization cycle to determinenew amplitudes t_(k)(2) and Bloch angles. By virtue of variationalprinciple, a new ground-state energy estimate E_(e)(2) thus obtained,will be no greater than E_(e)(1), namely,

E _(e)(2)≤E _(e)(1)  (7)

and closer to the true ground-state energy, which may improvedescription of a molecular system under study.

A method of performing an Iterative Qubit Coupled Cluster (iQCC)procedure may be described by the following operations. At an operation(1), the original electronic Hamiltonian H_(e) may be implemented as thefirst-step operator H_(e)(1). At an operation (2), a first entanglerT(1) may be identified. The first entangler may be identified accordingto QCC screening procedure described elsewhere herein. The firstentangler may be the entangler which most minimizes an eigenvalue of theHamiltonian to a threshold condition. At an operation (3), QCCcalculations may be performed to determine a value of the correspondingamplitude t(1). The value of the amplitude t(1) may be a value whichminimizes an eigenvalue of the Hamiltonian to a threshold condition. Atan operation (4), T(1) and t(1) may be inserted into Eq. (5) to dressH_(e)(1). This may result in the transformed Hamiltonian H_(e)(2). Atthis step, T(1) and t(1) may not be minimized while a value of T(2) andt(1) is determined. At an operation (5), operations (1) to (4) may berepeated. Operations (1) to (4) may be repeated until the ground-stateenergy estimate E_(e)(N) at the iteration N is found to be close to thevalue from the previous iteration,

E _(e)(N−1)−E _(e)(N)<ε

where ε is a threshold condition. In some cases, operations (1) to (4)may be repeated until the Hamiltonian H_(e)(N) becomes intractable, forexample, the dressing procedure may be too long for measurement processon a quantum hardware. The net result of such a procedure may be acircuit which requires a small number of quantum circuit gate operationsand which reduces error.

Implementation of the Quantum Mean Field Ansatz on the Quantum Computer

On a quantum computer, an initial estimate of the angle of each qubitmay be provided. Next, the interaction terms may be provided based on acalculation of the one and two electron integrals in the trialHamiltonian and the quantum computer initialized. This procedure maycomprise, for example, use of a variational quantum eigensolver or phaseestimation; however, any procedure for preparing trial states may beused. In some cases, embedding the Hamiltonian on the quantum computermay comprise generating the measurement topology which would yield thedistribution of measurements characteristic of the solutions to theHamiltonian. For example, in a universal gate quantum computer, theembedding may comprise generating a set of one qubit rotations (e.g. X,Ygates) to rotate the qubits into a trial state before measurement. Therotation may give the quantum computer its characteristic distributionof measurements. For example, in an annealer, the Hamiltonian may bephysically embedded (e.g. as qubit coupling and bias), whereas in auniversal gate, the Hamiltonian may be used to generate the measurementtopology.

After each trial state is prepared, the associated energy may beestimated by measuring and summing the energy of the individual Pauliterms in the Hamiltonian. The energy estimates may then be used in agradient descent algorithm to optimize the control parameters. Thecontrol parameter may be minimization of the energy of a quantum state.Additionally or alternatively, the control parameter may be an operatorthat commutes with the Hamiltonian, as described herein, the above stepsmay then be repeated until the control parameter is sufficientlyminimized. Sufficient minimization may comprise when changes in thecontrol parameter are below a threshold value. Additionally oralternatively, sufficient minimization may comprise having reached apredetermined number of iterations. Additionally or alternatively,sufficient minimization may comprise having reached a predeterminedcalculation time, such as a time at which the qubits have lostcoherence.

At an operation 275 of the method 200, the calculation may be embeddedon a quantum computer. FIG. 5 and FIG. 6 may comprise embodiments,variations, and examples of operation 275 of the method 200.

FIG. 5 shows an example implementation of a method 500 of solving aproblem using the quantum mean-field ansatz. At an operation 510 of themethod 500, a Hamiltonian in qubit space may be provided. At anoperation 520 of the method 500, a quantum mean field ansatz may beprovided. At an operation 530 of the method 500, the circuits of thequantum computer may be finalized using a variational quantumeigensolver or phase estimation and may be embedded onto the quantumcomputer with an estimate of the initial angles. An initial estimate ofthe angles may be made on the classical computer. An initial estimate ofthe angles may be provided to the quantum computer by the classicalcomputer. At an operation 540 of the method 500 the expectation valuefor each term of the Hamiltonian may be measured and summed together.Operations 530 and 540 may be repeated at an operation 550 until acriterion is reached. The criterion may comprise a threshold value. Thecriterion may comprise a minimum. The minimum may be a local minimum ora global minimum. The minimum may comprise a solution to the problem.

FIG. 6 shows a second example implementation of a method 600 of solvinga problem using the quantum mean-field ansatz. At an operation 610 ofthe method 600, a mean field Hamiltonian in qubit space may be provided.At an operation 620 of the method 600, a quantum mean field ansatz maybe provided. At an operation 630 of the method 600 an entangling schemebased on alpha-beta MO blending schemes may be implemented. At anoperation 640 of the method 600, the circuits of the quantum computermay be finalized using a variational quantum eigensolver or phaseestimation and may be embedded onto the quantum computer with anestimate of the initial angles. An initial estimate of the angles may bemade on the classical computer. An initial estimate of the angles may beprovided to the quantum computer by the classical computer. At anoperation 650 of the method 600 the expectation value for each term inthe Hamiltonian may be measured and summed together. Operations 630,640, and 650 may be repeated at an operation 660 until a criterion isreached. The criterion may comprise a threshold value. The criterion maycomprise a minimum. The minimum may be a local minimum or a globalminimum. The minimum may comprise a solution to the problem.

EXAMPLES

FIG. 7 shows the two lowest eigenstates of the Hamiltonian for H₂, theminimum of the corresponding quantum mean-field (QMF) functional, andexperimental data from a Rigetti quantum computer. The two lowesteigenstates of the Hamiltonian correspond to the exact potential energycurves calculated by full diagonalization of the qubit Hamiltonian fordifferent R. The QMF solution was obtained by minimizing the QMF energywith respect to all 8 Bloch angles using the sequential quadraticprograming algorithm s implemented by the fmincon routine in the MATLABsoftware. Calculation were performed using a STO-3G basis set mapped to4 qubits using the Bravyi-Kitaev transformation. The resultingHamiltonian has 15 terms, each of which is inferred from one- andtwo-electron molecular integrals at a given interatomic distance R.

FIG. 8 shows the two lowest eigenstates of the Hamiltonian for H₂, theminimum of the corresponding QMF functional, and experimental data froma Rigetti quantum computer, where the QMF functional has beenconstrained by S²=0. FIG. 8 shows the results of constrainedminimization of EQMF together with the exact curves shown in FIG. 7. Theconstrained mean-field potential energy curve is smooth and retains thenumber and/or spin characteristic over all R. The constrained mean-fieldpotential additionally retains the same behavior as the RHF curve givenby traditional quantum chemistry in so far as it produces the incorrectdissociation limit that is half way between the purely radical H+H andionic H⁺+H⁻ dissociation limit. Without the constraints, it may be verydifficult to reach the ground state of the electronic structure becauseother states aside from the one of interest (such as A+ and A− for anarbitrary molecule A) may be embedded into the VQE solution. This cancause the VQE ground state to be inaccurate relative to the actualground state of a given configuration, dependent on number of electronsand the spin (N, S²).

FIG. 9 shows the two lowest eigenstates of the Hamiltonian for H₂ ⁺, theminimum of the corresponding QMF functional, and experimental data froma Rigetti quantum computer, where the QMF functional has beenconstrained by N=1. Similar to FIG. 8, the constrained minimization canalso be employed to extract the potential energy curve of H₂ ⁺. Theresulting curve is smooth and actually follows the H₂ ⁺ potential energycurve computed by the GAMES S quantum chemistry package with the sameSTO-3G basis set.

FIG. 11 shows correlated mean-field energy E_(cQMF) ^(x) ¹ ^(z) ⁰ (τ)for the generator {circumflex over (T)}₁=x₁z₀ for H₂ in a minimal basisusing the coupled cluster method described herein. FIG. 11 shows theexact energy E₀ and E_(QMF) for reference. For the example generator,the correlated energy is lower than E_(QMF) everywhere and reaches theexact value at τ=π/4. In an appropriate basis, such as shown thisexample, the problem can be solved by a single qubit rotation.

FIG. 12 shows correlated mean-field energy E_(cQMF) ^(x) ¹ ^(x) ⁰ (τ)for the generator {circumflex over (T)}₂=x₁x₀ for H₂ in a minimal basisusing the coupled cluster method described herein. The entangler in thisexample, never goes below E_(QMF), therefore, this entangler may be lessuseful than {circumflex over (T)}₁=x₁z₀. Any fixed-amplitude entanglerwith this generator worsens the accuracy of the variational procedure.By The previous two examples show that with appropriate choice ofentangler, 4 Bloch angles and one τ amplitude may be used to achieve theexact solutions (5 qubits). In the procedure of Kandala et al. 10optimization parameters n(3d+2) for n=2 and d=1 were used for the sameHamiltonian.

FIG. 15 shows a calculated potential energy curve for the bondstretching coordinate of the H2 molecule using the example QCCentanglement ansatz of FIG. 14. For comparison, FIG. 15 also showsresults for the QMF method, spin constrained QMF method, and the exactpotential energy surface. In the H2 example, the qubit/spin Hamiltonianacts upon 4 qubits. In other embodiments the Hamiltonian may be reducedthrough parity mapping and by eliminating stationary qubits (i.e. qubitsthat have Pauli Z or identity operators on them, which corresponds to aneigenvalue of ±1). In this example, 54 two qubit combinations of variousPauli spin operators were tested, where six Pauli word entanglers—x₂y₀,y₂x₀, z₂y₀, z₂x₀, y₂z₀, x₂z₀, were used in the QCC Hamiltonian, sincethese six lower the energy of the molecule. In this particular example,one entanglement gate of these Pauli operators is needed, since they areall related by a simple global frame rotation. In this example theentangler P=x₂y₀ is used which produces the FCI result on the PES for H₂shown in FIG. 15. The ranking of the two qubit entanglers for the H2molecule is shown in Table 1.

TABLE 1 Entangler P$ \frac{{dE}\lbrack {\tau;} \rbrack}{d\;\tau} |_{\tau = 0}$$ \frac{d^{2}{E\lbrack {\tau;} \rbrack}}{d\tau^{2}} |_{\tau = 0}$ΔE[

] x₂y₀ −0.3936 0.5351 −0.0350 y₂x₀ 0.3939 0.5351 −0.0350 z₂y₀ 0.000−0.0502 −0.0350 z₂x₀ 0.000 −0.0502 −0.0350 y₂z₀ 0.000 −0.0493 −0.0350x₂z₀ 0.000 −0.0493 −0.0350

FIG. 17 shows a calculated potential energy curve for the bondstretching coordinate of LiH using the QCC entanglement scheme describedherein. For comparison, FIG. 17 also shows results for the QMF method,spin constrained QMF method, and the exact potential energy surface. Inthe LiH example, there are 2 two-qubit, 12 three qubit, and 18four-qubit entanglers. They can be reduced by combining them into groupswith the same absolute value of the gradient. The list was truncated toone two-qubit and 2 four-qubit mutually commuting entanglers with thelargest magnitude and the same sign of the energy gradient (x₂y₀,x₂x₃x₁y₀, and y₃x₂y₁y₀). As shown in Table 2, at R=1.5 angstroms bothfour-qubit entanglers are an order of magnitude more energeticallyimportant than the two-qubit one. Before factorization, there are 11variational parameters in total, N_(q)=4 and N_(ent)=3. Subsequently,the four-qubit entanglers were factored into 7 two-qubit entanglers. Thenumber of variational parameters does not grow using this method.Sub-milliHartree accuracy for the whole range of R(Li—H) was reachedwith 7 multi-qubit entanglers.

TABLE 2 Entangler P$ \frac{{dE}\lbrack {\tau;} \rbrack}{d\;\tau} |_{\tau = 0}$$ \frac{d^{2}{E\lbrack {\tau;} \rbrack}}{d\;\tau^{2}} |_{\tau = 0}$ΔE[

] x₂y₀ 0.0000 −0.0121 −0.0000 x₂x₃x₁y₀ −0.0653 0.1742 −0.0025 y₃x₂y₁y₀−0.0401 −0.2783 −0.0003 z₃z₂y₁x₀ −0.0156 −4.5886 −0.0067 x₃x₂y₁ 0.0072−1.2785 −0.0003 y₃z₂x₀ −0.0626 −15.3987 −0.0539 x₃z₂y₁x₀ −0.0249 −3.2103−0.0009

FIG. 20 shows the results of implementing the QCC method without domainfolding (2010)—solved exclusively on a classical computer—and withdomain folding (2020) solving the discrete optimization on a quantumannealer of LiH for a bond distance of 1.45 A and each condition wasrepeated 101 times.

Classical Computer Control Systems

The present disclosure provides classical computer control systems thatare programmed to implement methods of the disclosure. FIG. 10 shows aclassical computer system 1001 that is programmed or otherwiseconfigured to control operation of the quantum computer or a quantumprocessing unit within the quantum computer. The computer system 1001can regulate various aspects of the quantum computer 1002 of the presentdisclosure, such as, for example, implementing a method of solving aproblem or a method of solving an electronic structure problem describedherein. The computer system 1001 may perform one or more classicalcomputations, which may comprise precursor or intermediate orpost-processing steps to the methods described herein. The computersystem may aid or assist in embedding one or more parts of theHamiltonian the quantum computer 1002. The classical computer system1001 may set and/or determine the bias and/or the coupling each qubit.Any operation of the methods and systems described herein which do notdirectly depend on the quantum nature of the qubit may be delegated to aclassical computer as necessary. In some cases, the computer system 1001can simulate a quantum computer.

The computer system 1001 can be an electronic device of a user or acomputer system that is remotely located with respect to the electronicdevice. The electronic device can be a mobile electronic device. Thecomputer system 1001 includes a central processing unit (CPU, also“processor” and “computer processor” herein) 1005, which can be a singlecore or multi core processor, or a plurality of processors for parallelprocessing. The computer system 1001 also includes memory or memorylocation 1010 (e.g., random-access memory, read-only memory, flashmemory), electronic storage unit 1015 (e.g., hard disk), communicationinterface 1020 (e.g., network adapter) for communicating with one ormore other systems, and peripheral devices 1025, such as cache, othermemory, data storage and/or electronic display adapters. The memory1010, storage unit 1015, interface 1020 and peripheral devices 1025 arein communication with the CPU 1005 through a communication bus (solidlines), such as a motherboard. The storage unit 1015 can be a datastorage unit (or data repository) for storing data. The computer system1001 can be operatively coupled to a computer network (“network”) 1030with the aid of the communication interface 1020. The network 1030 canbe the Internet, an internet and/or extranet, or an intranet and/orextranet that is in communication with the Internet. The network 1030 insome cases is a telecommunication and/or data network. The network 1030can include one or more computer servers, which can enable distributedcomputing, such as cloud computing. The network 1030, in some cases withthe aid of the computer system 1001, can implement a peer-to-peernetwork, which may enable devices coupled to the computer system 1001 tobehave as a client or a server.

The CPU 1005 can execute a sequence of machine-readable instructions,which can be embodied in a program or software. The instructions may bestored in a memory location, such as the memory 1010. The instructionscan be directed to the CPU 1005, which can subsequently program orotherwise configure the CPU 1005 to implement methods of the presentdisclosure. Examples of operations performed by the CPU 1005 can includefetch, decode, execute, and writeback.

The CPU 1005 can be part of a circuit, such as an integrated circuit.One or more other components of the system 1001 can be included in thecircuit. In some cases, the circuit is an application specificintegrated circuit (ASIC).

The classical storage unit 1015 can store files, such as drivers,libraries and saved programs. The storage unit 1015 can store user data,e.g., user preferences and user programs. The computer system 1001 insome cases can include one or more additional data storage units thatare external to the computer system 1001, such as located on a remoteserver that is in communication with the computer system 1001 through anintranet or the Internet.

The computer system 1001 can communicate with one or more remotecomputer systems through the network 1030. For instance, the computersystem 1001 can communicate with a remote computer system of a user.Examples of remote computer systems include personal computers (e.g.,portable PC), slate or tablet PC's (e.g., Apple® iPad, Samsung® GalaxyTab), telephones, Smart phones (e.g., Apple® iPhone, Android-enableddevice, Blackberry®), or personal digital assistants. The user canaccess the computer system 1001 and thus the quantum computer system1002 via the network 1030.

Methods as described herein can be implemented by way of machine (e.g.,computer processor) executable code stored on an electronic storagelocation of the computer system 1001, such as, for example, on thememory 1010 or classical storage unit 1015. The machine executable ormachine readable code can be provided in the form of software. Duringuse, the code can be executed by the processor 1005. In some cases, thecode can be retrieved from the storage unit 1015 and stored on thememory 1010 for ready access by the processor 1005. In some situations,the electronic storage unit 1015 can be precluded, andmachine-executable instructions are stored on memory 1010.

The code can be pre-compiled and configured for use with a machinehaving a processer adapted to execute the code, or can be compiledduring runtime. The code can be supplied in a programming language thatcan be selected to enable the code to execute in a pre-compiled oras-compiled fashion.

Aspects of the systems and methods provided herein, such as the computersystem 1001, can be embodied in programming. Various aspects of thetechnology may be thought of as “products” or “articles of manufacture”typically in the form of machine (or processor) executable code and/orassociated data that is carried on or embodied in a type of machinereadable medium. Machine-executable code can be stored on an electronicstorage unit, such as memory (e.g., read-only memory, random-accessmemory, flash memory) or a hard disk. “Storage” type media can includeany or all of the tangible memory of the computers, processors or thelike, or associated modules thereof, such as various semiconductormemories, tape drives, disk drives and the like, which may providenon-transitory storage at any time for the software programming. All orportions of the software may at times be communicated through theInternet or various other telecommunication networks. Suchcommunications, for example, may enable loading of the software from onecomputer or processor into another, for example, from a managementserver or host computer into the computer platform of an applicationserver. Thus, another type of media that may bear the software elementsincludes optical, electrical and electromagnetic waves, such as usedacross physical interfaces between local devices, through wired andoptical landline networks and over various air-links. The physicalelements that carry such waves, such as wired or wireless links, opticallinks or the like, also may be considered as media bearing the software.As used herein, unless restricted to non-transitory, tangible “storage”media, terms such as computer or machine “readable medium” refer to anymedium that participates in providing instructions to a processor forexecution.

Hence, a machine readable medium, such as computer-executable code, maytake many forms, including but not limited to, a tangible storagemedium, a carrier wave medium or physical transmission medium.Non-volatile storage media include, for example, optical or magneticdisks, such as any of the storage devices in any computer(s) or thelike, such as may be used to implement the databases, etc. shown in thedrawings. Volatile storage media include dynamic memory, such as mainmemory of such a computer platform. Tangible transmission media includecoaxial cables; copper wire and fiber optics, including the wires thatcomprise a bus within a computer system. Carrier-wave transmission mediamay take the form of electric or electromagnetic signals, or acoustic orlight waves such as those generated during radio frequency (RF) andinfrared (IR) data communications. Common forms of computer-readablemedia therefore include for example: a floppy disk, a flexible disk,hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD orDVD-ROM, any other optical medium, punch cards paper tape, any otherphysical storage medium with patterns of holes, a RAM, a ROM, a PROM andEPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wavetransporting data or instructions, cables or links transporting such acarrier wave, or any other medium from which a computer may readprogramming code and/or data. Many of these forms of computer readablemedia may be involved in carrying one or more sequences of one or moreinstructions to a processor for execution.

The computer system 1001 can include or be in communication with anelectronic display 1035 that comprises a user interface (UI) 1040 forproviding, for example, results or solutions to the problems describedherein. Examples of UI's include, without limitation, a graphical userinterface (GUI) and web-based user interface.

The classical computer system may be operably connected to a quantumcomputer system 1002. The quantum computer system may comprise a quantumprocessing unit 1006, which may further comprise qubits. A quantumcomputer may comprise a number of qubits which number may be forexample, 1, 2, 5, 10, 20, 50, 100, 1,000, 10,000, 100,000, 1 million, 1billion, 1 trillion, or any number of qubits defined by a range betweenany two of the preceding values.

The quantum processing unit 1006 may execute a sequence of instructions,which can be embodied in a program or software. The instructions may bestored in a memory location, such as the memory 1010 of classicalcomputer system 1001. The instructions can be directed to the quantumprocessing unit 1006, which can subsequently program or otherwiseconfigure the quantum processing unit 1006 to implement methods of thepresent disclosure. The quantum processing unit 1006 can be part of acircuit, such as a quantum logic circuit. One or more other componentsof the systems 1001 and 1002 can be included in the circuit.

Quantum computer system 1002 may comprise a quantum storage unit 1016.The quantum storage unit may be configured to store quantum information.The quantum storage unit may comprise additional qubits.

While preferred embodiments of the present invention have been shown anddescribed herein, it will be obvious to those skilled in the art thatsuch embodiments are provided by way of example only. Numerousvariations, changes, and substitutions will now occur to those skilledin the art without departing from the invention. It should be understoodthat various alternatives to the embodiments of the invention describedherein may be employed in practicing the invention. It is intended thatthe following claims define the scope of the invention and that methodsand structures within the scope of these claims and their equivalents becovered thereby.

What is claimed is:
 1. (canceled)
 2. (canceled)
 3. (canceled) 4.(canceled)
 5. The method of claim 107, wherein the parameterization ofthe qubit Hamiltonian in Pauli Z rotations comprises an Ising-typeHamiltonian.
 6. The method of claim 107, wherein the parameterization ofthe qubit Hamiltonian in Pauli Z rotations comprises a higher orderbinary optimization (HOBO) problem.
 7. (canceled)
 8. (canceled) 9.(canceled)
 10. (canceled)
 11. (canceled)
 12. (canceled)
 13. (canceled)14. (canceled)
 15. The method of claim 78, wherein the qubit Hamiltonianis a mean-field Hamiltonian.
 16. The method of claim 78, wherein thequbit Hamiltonian comprises a transformation of a fermionic Hamiltonian.17. (canceled)
 18. (canceled)
 19. (canceled)
 20. (canceled) 21.(canceled)
 22. (canceled)
 23. (canceled)
 24. (canceled)
 25. (canceled)26. (canceled)
 27. (canceled)
 28. (canceled)
 29. (canceled) 30.(canceled)
 31. The method of claim 78, wherein the action of embeddingcomprises transferring a value of at least one term selected from a biasterm and a coupling term to the quantum computer.
 32. (canceled) 33.(canceled)
 34. (canceled)
 35. (canceled)
 36. (canceled)
 37. (canceled)38. (canceled)
 39. The method of claim 78, wherein the action ofidentifying comprises transforming a fermionic Hamiltonian to the qubitHamiltonian using at least one of a Jordan-Wigner transformation, theBravyi-Kitaev method and the Parity method.
 40. (canceled) 41.(canceled)
 42. (canceled)
 43. The method of claim 142, wherein a numberof instances used of operating at least one quantum logic gate scaleslinearly with a number of fermions in the qubit Hamiltonian. 44.(canceled)
 45. (canceled)
 46. (canceled)
 47. (canceled)
 48. (canceled)49. (canceled)
 50. (canceled)
 51. (canceled)
 52. (canceled)
 53. Aquantum computer comprising: a plurality of qubits; a qubit-coupledcluster Hamiltonian embedded on the quantum computer, wherein at leastone eigenvalue thereof is a variational upper bound to an exact stateenergy; and at least one quantum logic gate operation for implementationon the quantum computer, comprising: at least one entangler comprisingat least one Pauli Word; and a selected one of the at least oneentangler comprising selected at least ones of the Pauli Words thatreduce a value of the at least one eigenvalues.
 54. (canceled) 55.(canceled)
 56. (canceled)
 57. (canceled)
 58. (canceled)
 59. (canceled)60. (canceled)
 61. (canceled)
 62. (canceled)
 63. (canceled) 64.(canceled)
 65. (canceled)
 66. (canceled)
 67. (canceled)
 68. (canceled)69. The method of claim 143, wherein the at least one operator comprisesa commutation relation of the qubit Hamiltonian.
 70. The method of claim143, wherein the at least one operator is at least one of a numberoperator and a total spin operator.
 71. (canceled)
 72. (canceled) 73.(canceled)
 74. (canceled)
 75. (canceled)
 76. (canceled)
 77. (canceled)78. A method of solving a problem on a quantum computer, the methodcomprising actions of: identifying a qubit Hamiltonian parameterized byat least one entangler and having at least one eigenvalue, wherein theat least one entangler is ranked by a first order energy derivative andthe at least one eigenvalue comprises a variational upper bound to anexact state energy; selecting a qubit-coupled cluster Hamiltoniancomprising a selected one of the at least one entangler that reduces avalue of a selected one of the at least one eigenvalue; embedding theselected qubit-coupled cluster Hamiltonian on the quantum computer tooptimize an amplitude of the selected entangler that reduces the valueof the selected one of the at least one eigenvalue; evaluating theamplitude of the selected entangler to select a different qubit-coupledcluster Hamiltonian comprising a different selected one of the at leastone entangler that reduces a value of a different selected one of the atleast one eigenvalue; and repeating the actions of embedding andevaluating until a solution is obtained.
 79. The method of claim 0,wherein the action of selecting comprises parameterizing at least onecoordinate of the qubit-coupled cluster Hamiltonian by a spin coherentstate.
 80. The method of claim 0, further comprising folding anoptimization domain in at least one of an amplitude space and a phasespace before the action of selecting.
 81. (canceled)
 82. The method ofclaim 78, wherein the action of selecting comprises applying thefollowing to transform the qubit-Hamiltonian into the qubit-coupledcluster Hamiltonian:H _(e)+sin(t ₁)/2(T ₁ H _(e) −H _(e) T ₁)+(1−cos(t ₁))/2(T ₁ H ₁ −H_(e)).
 83. (canceled)
 84. (canceled)
 85. (canceled)
 86. The method ofclaim 78, wherein a solution to the problem comprises reduced values ofthe at least one selected eigenvalues.
 87. (canceled)
 88. The method ofclaim 0, wherein the solution is at least one of a global minimum and alocal minimum.
 89. (canceled)
 90. (canceled)
 91. (canceled) 92.(canceled)
 93. (canceled)
 94. (canceled)
 95. (canceled)
 96. (canceled)97. A computer, comprising: a processor; a machine-readable,non-transitory storage medium for storing instructions that, whenexecuted by the processor, cause the computer to perform a method ofsolving a problem, comprising actions of: identifying a qubitHamiltonian parameterized by at least one entangler and having at leastone eigenvalue, wherein the at least one entangler is ranked by a firstorder energy derivative and the at least one eigenvalue comprises avariational upper bound to an exact state energy, wherein one or morecoordinates in the qubit Hamiltonian comprises a parameterization in aspin coherent state, selecting a qubit-coupled cluster Hamiltoniancomprising a selected one of the at least one entangler that reduces avalue of a selected one of the at least one eigenvalue; embedding theselected qubit-coupled cluster Hamiltonian on a quantum computer tooptimize an amplitude of the selected entangler that reduces the valueof the selected one of the at least one eigenvalue; evaluating theamplitude of the selected entangler to select a different qubit-coupledcluster Hamiltonian comprising a different selected one of the at leastone entangler that reduces a value of a different selected one of the atleast one eigenvalue; and repeating the actions of embedding andevaluating until a solution is obtained.
 98. The computer of claim 97,wherein the processor is a classical computer.
 99. The computer of claim98, wherein the processor simulates the operation of the quantumcomputer claim
 53. 100. (canceled)
 101. (canceled)
 102. (canceled) 103.The method of claim 79, wherein the action of parameterizing by a spincoherent state comprises an expression in spherical polar coordinates ona Bloch sphere.
 104. The method of claim 79, wherein the spin coherentstate is parameterized by the following:$ {{{ {{\mspace{20mu}{ \Omega \rangle = {{\cos^{2J}( \frac{\theta}{2} )}{\exp\lbrack {{\tan( \frac{\theta}{2} )}e^{i\;\phi}{\hat{S}}_{-}} \rbrack}}}}{JJ}} \rangle,{ \Omega \rangle = {\sum_{M = {- J}}^{J}{\begin{pmatrix}{2J} \\{M + J}\end{pmatrix}^{1/2} \times {\cos^{J + M}( \frac{\theta}{2} )}{\sin^{J - M}( \frac{\theta}{2} )}e^{{i{({J - M})}}\phi}}}}}}{JM}} \rangle.$105. (canceled)
 106. (canceled)
 107. The method of claim 78, wherein thequbit Hamiltonian is parameterized in Pauli Z rotations.
 108. The methodof claim 78, wherein the qubit Hamiltonian comprises at least one of aquadratic unconstrained boundary optimization (QUBO) model, k-localIsing model, higher order binary optimization (HOBO) model, and anycombination of any of these.
 109. The method of claim 107, wherein thequbit Hamiltonian parameterized in Pauli Z rotations comprises at leastone term selected from a bias term and a coupling term.
 110. (canceled)111. (canceled)
 112. (canceled)
 113. (canceled)
 114. The method of claim78, wherein the qubit-coupled cluster Hamiltonian comprises at least oneentangler parameterized by a spin coherent state.
 115. The method ofclaim 114, wherein the at least one entangler comprises at least onePauli Word.
 116. The method of claim 115, wherein the selected one ofthe at least one entangler comprises selected Pauli Words that reduce avalue of the at least one eigenvalue.
 117. The method of claim 116,wherein the selected one of the at least one entangler is expressed asPauli Z rotations.
 118. The method of claim 78, wherein an optimizationdomain of the qubit Hamiltonian is folded in at least one of amplitudeand phase.
 119. A computer-implemented method of solving a problemcomprising: providing a qubit Hamiltonian, wherein at least oneeigenvalue thereof is a variational upper bound to an exact stateenergy; parameterizing at least one coordinate in the qubit Hamiltonianby a spin coherent state; and providing a solution to the problem thatcomprises at least one of the at least one eigenvalue of the qubitHamiltonian.
 120. (canceled)
 121. The method of claim 119, furthercomprising directing the qubit Hamiltonian to be embedded on a quantumcomputer.
 122. The method of claim 121, further comprising receiving thesolution from the quantum computer.
 123. The method of claim 121,wherein the quantum computer is simulated on a classical computer. 124.The method of claim 119, wherein the action of parameterizing comprisesexpressing the spin coherent state in spherical polar coordinates on aBloch sphere.
 125. The method of claim 119, wherein the action ofparameterizing in a spin coherent state comprises applying:$ {{{ {{\mspace{20mu}{ \Omega \rangle = {{\cos^{2J}( \frac{\theta}{2} )}{\exp\lbrack {{\tan( \frac{\theta}{2} )}e^{i\;\phi}{\hat{S}}_{-}} \rbrack}}}}{JJ}} \rangle,{ \Omega \rangle = {\sum_{M = {- J}}^{J}{\begin{pmatrix}{2J} \\{M + J}\end{pmatrix}^{1/2} \times {\cos^{J + M}( \frac{\theta}{2} )}{\sin^{J - M}( \frac{\theta}{2} )}e^{{i{({J - M})}}\phi}}}}}}{JM}} \rangle.$126. The method of claim 119, wherein the solution is at least one of aglobal minimum and a local minimum.
 127. (canceled)
 128. The method ofclaim 119, wherein the action of parameterizing comprises expressing thequbit Hamiltonian in Pauli Z rotations.
 129. The method of claim 119,wherein the action of parameterizing comprises employing a quadraticunconstrained boundary optimization (QUBO) model, k-local Ising model,higher order binary optimization (HOBO) model, and any combination ofany of these.
 130. The method of claim 119, wherein the action ofparameterizing comprises at least one quantum logic gate.
 131. Themethod of claim 130, further comprising transforming the qubitHamiltonian into a qubit-coupled cluster Hamiltonian.
 132. The method ofclaim 131, wherein the action of transforming comprises applying thefollowing:H _(e)+sin(t ₁)/2(T ₁ H _(e) −H _(e) T ₁)+(1−cos(t ₁))/2(T ₁ H _(e) T ₁−H _(e)).
 133. The method of claim 131, wherein the qubit-coupledcluster Hamiltonian comprises at least one entangler parameterized by aspin coherent state.
 134. The method of claim 131, wherein the at leastone entangler comprises at least one Pauli Words.
 135. The method ofclaim 134, wherein the action of transforming comprises selecting PauliWords that reduce a value of the at least one eigenvalue to select asubset of the at least one entangler.
 136. The method of claim 135,wherein the subset of the at least one entangler is expressed as Pauli Zrotations.
 137. The method of claim 119, further comprising folding anoptimization domain of the qubit Hamiltonian in at least one of anamplitude space and a phase space.
 138. A computer comprising: aprocessor; a machine-readable, non-transitory storage medium for storinginstructions that, when executed by the computer, cause the computer toperform a method of solving a problem, comprising actions of: providinga qubit Hamiltonian, wherein at least one eigenvalue thereof is avariational upper bound to an exact stage energy; parameterizing atleast one coordinate in the qubit Hamiltonian by a spin coherent state;and providing a solution to the problem that comprises at least one ofthe at least one eigenvalue of the qubit Hamiltonian.
 139. The system ofclaim 6, wherein the HOBO problem comprises a quadratic unconstrainedboundary optimization (QUBO).
 140. The method of claim 79, wherein theaction of parameterizing comprises expressing the qubit-coupled clusterHamiltonian in Pauli Z rotations.
 141. The method of claim 121, whereinthe quantum computer is at least one of a quantum annealer and auniversal gate quantum computing unit.
 142. The method of claim 114,wherein the parameterization by a spin coherent state comprisesoperating at least one quantum logic gate.
 143. The method of claim 78,further comprising an action of providing at least one operator thatcommutes with the qubit Hamiltonian.